Metallic ratios: Difference between revisions
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Golden
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
Silver
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Nickel
1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686
25 5.19258240356725201562535524577016
Aluminium
1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775
23 6.16227766016837933199889354443272
Iron
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Revision as of 11:28, 14 December 2022
You are encouraged to solve this task according to the task description, using any language you may know.
Many people have heard of the Golden ratio, phi (φ). Phi is just one of a series of related ratios that are referred to as the "Metallic ratios".
The Golden ratio was discovered and named by ancient civilizations as it was thought to be the most pure and beautiful (like Gold). The Silver ratio was was also known to the early Greeks, though was not named so until later as a nod to the Golden ratio to which it is closely related. The series has been extended to encompass all of the related ratios and was given the general name Metallic ratios (or Metallic means). Somewhat incongruously as the original Golden ratio referred to the adjective "golden" rather than the metal "gold".
Metallic ratios are the real roots of the general form equation:
x2 - bx - 1 = 0
where the integer b determines which specific one it is.
Using the quadratic equation:
( -b ± √(b2 - 4ac) ) / 2a = x
Substitute in (from the top equation) 1 for a, -1 for c, and recognising that -b is negated we get:
( b ± √(b2 + 4) ) ) / 2 = x
We only want the real root:
( b + √(b2 + 4) ) ) / 2 = x
When we set b to 1, we get an irrational number: the Golden ratio.
( 1 + √(12 + 4) ) / 2 = (1 + √5) / 2 = ~1.618033989...
With b set to 2, we get a different irrational number: the Silver ratio.
( 2 + √(22 + 4) ) / 2 = (2 + √8) / 2 = ~2.414213562...
When the ratio b is 3, it is commonly referred to as the Bronze ratio, 4 and 5 are sometimes called the Copper and Nickel ratios, though they aren't as standard. After that there isn't really any attempt at standardized names. They are given names here on this page, but consider the names fanciful rather than canonical.
Note that technically, b can be 0 for a "smaller" ratio than the Golden ratio. We will refer to it here as the Platinum ratio, though it is kind-of a degenerate case.
Metallic ratios where b > 0 are also defined by the irrational continued fractions:
[b;b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b...]
So, The first ten Metallic ratios are:
Metallic ratios Name b Equation Value Continued fraction OEIS link Platinum 0 (0 + √4) / 2 1 - - Golden 1 (1 + √5) / 2 1.618033988749895... [1;1,1,1,1,1,1,1,1,1,1...] OEIS:A001622 Silver 2 (2 + √8) / 2 2.414213562373095... [2;2,2,2,2,2,2,2,2,2,2...] OEIS:A014176 Bronze 3 (3 + √13) / 2 3.302775637731995... [3;3,3,3,3,3,3,3,3,3,3...] OEIS:A098316 Copper 4 (4 + √20) / 2 4.23606797749979... [4;4,4,4,4,4,4,4,4,4,4...] OEIS:A098317 Nickel 5 (5 + √29) / 2 5.192582403567252... [5;5,5,5,5,5,5,5,5,5,5...] OEIS:A098318 Aluminum 6 (6 + √40) / 2 6.16227766016838... [6;6,6,6,6,6,6,6,6,6,6...] OEIS:A176398 Iron 7 (7 + √53) / 2 7.140054944640259... [7;7,7,7,7,7,7,7,7,7,7...] OEIS:A176439 Tin 8 (8 + √68) / 2 8.123105625617661... [8;8,8,8,8,8,8,8,8,8,8...] OEIS:A176458 Lead 9 (9 + √85) / 2 9.109772228646444... [9;9,9,9,9,9,9,9,9,9,9...] OEIS:A176522
There are other ways to find the Metallic ratios; one, (the focus of this task)
is through successive approximations of Lucas sequences.
A traditional Lucas sequence is of the form:
xn = P * xn-1 - Q * xn-2
and starts with the first 2 values 0, 1.
For our purposes in this task, to find the metallic ratios we'll use the form:
xn = b * xn-1 + xn-2
( P is set to b and Q is set to -1. ) To avoid "divide by zero" issues we'll start the sequence with the first two terms 1, 1. The initial starting value has very little effect on the final ratio or convergence rate. Perhaps it would be more accurate to call it a Lucas-like sequence.
At any rate, when b = 1 we get:
xn = xn-1 + xn-2
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
more commonly known as the Fibonacci sequence.
When b = 2:
xn = 2 * xn-1 + xn-2
1, 1, 3, 7, 17, 41, 99, 239, 577, 1393...
And so on.
To find the ratio by successive approximations, divide the (n+1)th term by the
nth. As n grows larger, the ratio will approach the b metallic ratio.
For b = 1 (Fibonacci sequence):
1/1 = 1 2/1 = 2 3/2 = 1.5 5/3 = 1.666667 8/5 = 1.6 13/8 = 1.625 21/13 = 1.615385 34/21 = 1.619048 55/34 = 1.617647 89/55 = 1.618182 etc.
It converges, but pretty slowly. In fact, the Golden ratio has the slowest possible convergence for any irrational number.
- Task
For each of the first 10 Metallic ratios; b = 0 through 9:
- Generate the corresponding "Lucas" sequence.
- Show here, on this page, at least the first 15 elements of the "Lucas" sequence.
- Using successive approximations, calculate the value of the ratio accurate to 32 decimal places.
- Show the value of the approximation at the required accuracy.
- Show the value of n when the approximation reaches the required accuracy (How many iterations did it take?).
Optional, stretch goal - Show the value and number of iterations n, to approximate the Golden ratio to 256 decimal places.
You may assume that the approximation has been reached when the next iteration does not cause the value (to the desired places) to change.
- See also
C#
Since the task description outlines how the results can be calculated using square roots for each B, this program not only calculates the results by iteration (as specified), it also checks each result with an independent result calculated by the indicated square root (for each B).
using static System.Math;
using static System.Console;
using BI = System.Numerics.BigInteger;
class Program {
static BI IntSqRoot(BI v, BI res) { // res is the initial guess
BI term = 0, d = 0, dl = 1; while (dl != d) { term = v / res; res = (res + term) >> 1;
dl = d; d = term - res; } return term; }
static string doOne(int b, int digs) { // calculates result via square root, not iterations
int s = b * b + 4; BI g = (BI)(Sqrt((double)s) * Pow(10, ++digs)),
bs = IntSqRoot(s * BI.Parse('1' + new string('0', digs << 1)), g);
bs += b * BI.Parse('1' + new string('0', digs));
bs >>= 1; bs += 4; string st = bs.ToString();
return string.Format("{0}.{1}", st[0], st.Substring(1, --digs)); }
static string divIt(BI a, BI b, int digs) { // performs division
int al = a.ToString().Length, bl = b.ToString().Length;
a *= BI.Pow(10, ++digs << 1); b *= BI.Pow(10, digs);
string s = (a / b + 5).ToString(); return s[0] + "." + s.Substring(1, --digs); }
// custom formating
static string joined(BI[] x) { int[] wids = {1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13};
string res = ""; for (int i = 0; i < x.Length; i++) res +=
string.Format("{0," + (-wids[i]).ToString() + "} ", x[i]); return res; }
static void Main(string[] args) { // calculates and checks each "metal"
WriteLine("Metal B Sq.Rt Iters /---- 32 decimal place value ----\\ Matches Sq.Rt Calc");
int k; string lt, t = ""; BI n, nm1, on; for (int b = 0; b < 10; b++) {
BI[] lst = new BI[15]; lst[0] = lst[1] = 1;
for (int i = 2; i < 15; i++) lst[i] = b * lst[i - 1] + lst[i - 2];
// since all the iterations (except Pt) are > 15, continue iterating from the end of the list of 15
n = lst[14]; nm1 = lst[13]; k = 0; for (int j = 13; k == 0; j++) {
lt = t; if (lt == (t = divIt(n, nm1, 32))) k = b == 0 ? 1 : j;
on = n; n = b * n + nm1; nm1 = on; }
WriteLine("{0,4} {1} {2,2} {3, 2} {4} {5}\n{6,19} {7}", "Pt Au Ag CuSn Cu Ni Al Fe Sn Pb"
.Split(' ')[b], b, b * b + 4, k, t, t == doOne(b, 32), "", joined(lst)); }
// now calculate and check big one
n = nm1 =1; k = 0; for (int j = 1; k == 0; j++) {
lt = t; if (lt == (t = divIt(n, nm1, 256))) k = j;
on = n; n += nm1; nm1 = on; }
WriteLine("\nAu to 256 digits:"); WriteLine(t);
WriteLine("Iteration count: {0} Matched Sq.Rt Calc: {1}", k, t == doOne(1, 256)); }
}
- Output:
Metal B Sq.Rt Iters /---- 32 decimal place value ----\ Matches Sq.Rt Calc Pt 0 4 1 1.00000000000000000000000000000000 True 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Au 1 5 78 1.61803398874989484820458683436564 True 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 Ag 2 8 44 2.41421356237309504880168872420970 True 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 CuSn 3 13 34 3.30277563773199464655961063373525 True 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 Cu 4 20 28 4.23606797749978969640917366873128 True 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 Ni 5 29 25 5.19258240356725201562535524577016 True 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 Al 6 40 23 6.16227766016837933199889354443272 True 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 Fe 7 53 22 7.14005494464025913554865124576352 True 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 Sn 8 68 20 8.12310562561766054982140985597408 True 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 Pb 9 85 20 9.10977222864644365500113714088140 True 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 Au to 256 digits: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144 Iteration count: 616 Matched Sq.Rt Calc: True
Note: All the metals (except bronze) in this task are elements, so those symbols are used. Bronze is usually approx. 88% Cu and 12 % Sn, and so it is labeled CuSn here.
C++
#include <boost/multiprecision/cpp_dec_float.hpp>
#include <iostream>
const char* names[] = { "Platinum", "Golden", "Silver", "Bronze", "Copper", "Nickel", "Aluminium", "Iron", "Tin", "Lead" };
template<const uint N>
void lucas(ulong b) {
std::cout << "Lucas sequence for " << names[b] << " ratio, where b = " << b << ":\nFirst " << N << " elements: ";
auto x0 = 1L, x1 = 1L;
std::cout << x0 << ", " << x1;
for (auto i = 1u; i <= N - 1 - 1; i++) {
auto x2 = b * x1 + x0;
std::cout << ", " << x2;
x0 = x1;
x1 = x2;
}
std::cout << std::endl;
}
template<const ushort P>
void metallic(ulong b) {
using namespace boost::multiprecision;
using bfloat = number<cpp_dec_float<P+1>>;
bfloat x0(1), x1(1);
auto prev = bfloat(1).str(P+1);
for (auto i = 0u;;) {
i++;
bfloat x2(b * x1 + x0);
auto thiz = bfloat(x2 / x1).str(P+1);
if (prev == thiz) {
std::cout << "Value after " << i << " iteration" << (i == 1 ? ": " : "s: ") << thiz << std::endl << std::endl;
break;
}
prev = thiz;
x0 = x1;
x1 = x2;
}
}
int main() {
for (auto b = 0L; b < 10L; b++) {
lucas<15>(b);
metallic<32>(b);
}
std::cout << "Golden ratio, where b = 1:" << std::endl;
metallic<256>(1);
return 0;
}
- Output:
Lucas sequence for Platinum ratio, where b = 0: First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Value after 1 iteration: 1 Lucas sequence for Golden ratio, where b = 1: First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 Value after 78 iterations: 1.61803398874989484820458683436564 Lucas sequence for Silver ratio, where b = 2: First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243 Value after 44 iterations: 2.4142135623730950488016887242097 Lucas sequence for Bronze ratio, where b = 3: First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564 Value after 34 iterations: 3.30277563773199464655961063373525 Lucas sequence for Copper ratio, where b = 4: First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141 Value after 28 iterations: 4.23606797749978969640917366873128 Lucas sequence for Nickel ratio, where b = 5: First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686 Value after 25 iterations: 5.19258240356725201562535524577016 Lucas sequence for Aluminium ratio, where b = 6: First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775 Value after 23 iterations: 6.16227766016837933199889354443272 Lucas sequence for Iron ratio, where b = 7: First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408 Value after 22 iterations: 7.14005494464025913554865124576352 Lucas sequence for Tin ratio, where b = 8: First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449 Value after 20 iterations: 8.12310562561766054982140985597408 Lucas sequence for Lead ratio, where b = 9: First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226 Value after 20 iterations: 9.1097722286464436550011371408814 Golden ratio, where b = 1: Value after 615 iterations: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
F#
// Metallic Ration. Nigel Galloway: September 16th., 2020
let rec fN i g (e,l)=match i with 0->g |_->fN (i-1) (int(l/e)::g) (e,(l%e)*10I)
let fI(P:int)=Seq.unfold(fun(n,g)->Some(g,((bigint P)*n+g,n)))(1I,1I)
let fG fI fN=let _,(n,g)=fI|>Seq.pairwise|>Seq.mapi(fun n g->(n,fN g))|>Seq.pairwise|>Seq.find(fun((_,n),(_,g))->n=g) in (n,List.rev g)
let mR n g=printf "First 15 elements when P = %d -> " n; fI n|>Seq.take 15|>Seq.iter(printf "%A "); printf "\n%d decimal places " g
let Σ,n=fG(fI n)(fN (g+1) []) in printf "required %d iterations -> %d." Σ n.Head; List.iter(printf "%d")n.Tail ;printfn ""
[0..9]|>Seq.iter(fun n->mR n 32; printfn ""); mR 1 256
- Output:
First 15 elements when P = 0 -> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 32 decimal places required 1 iterations -> 1.00000000000000000000000000000000 First 15 elements when P = 1 -> 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 32 decimal places required 79 iterations -> 1.61803398874989484820458683436563 First 15 elements when P = 2 -> 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 32 decimal places required 45 iterations -> 2.41421356237309504880168872420969 First 15 elements when P = 3 -> 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 32 decimal places required 33 iterations -> 3.30277563773199464655961063373524 First 15 elements when P = 4 -> 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 32 decimal places required 28 iterations -> 4.23606797749978969640917366873127 First 15 elements when P = 5 -> 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 32 decimal places required 25 iterations -> 5.19258240356725201562535524577016 First 15 elements when P = 6 -> 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 32 decimal places required 23 iterations -> 6.16227766016837933199889354443271 First 15 elements when P = 7 -> 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 32 decimal places required 21 iterations -> 7.14005494464025913554865124576351 First 15 elements when P = 8 -> 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 32 decimal places required 20 iterations -> 8.12310562561766054982140985597407 First 15 elements when P = 9 -> 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 32 decimal places required 19 iterations -> 9.10977222864644365500113714088139 First 15 elements when P = 1 -> 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 256 decimal places required 614 iterations -> 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
Factor
USING: combinators decimals formatting generalizations io kernel
math prettyprint qw sequences ;
IN: rosetta-code.metallic-ratios
: lucas ( n a b -- n a' b' ) tuck reach * + ;
: lucas. ( n -- )
1 pprint bl 1 1 14 [ lucas over pprint bl ] times 3drop nl ;
: approx ( a b -- d ) swap [ 0 <decimal> ] bi@ 32 D/ ;
: approximate ( n -- value iter )
-1 swap 1 1 0 1 [ 2dup = ]
[ [ 1 + ] 5 ndip [ lucas 2dup approx ] 2dip drop ] until
4nip decimal>ratio swap ;
qw{
Platinum Golden Silver Bronze Copper Nickel Aluminum Iron
Tin Lead
}
[
dup dup approximate {
[ "Lucas sequence for %s ratio " printf ]
[ "where b = %d:\n" printf ]
[ "First 15 elements: " write lucas. ]
[ "Approximated value: %.32f " printf ]
[ "- reached after %d iteration(s)\n\n" printf ]
} spread
] each-index
- Output:
Lucas sequence for Platinum ratio where b = 0: First 15 elements: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Approximated value: 1.00000000000000000000000000000000 - reached after 1 iteration(s) Lucas sequence for Golden ratio where b = 1: First 15 elements: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 Approximated value: 1.61803398874989484820458683436563 - reached after 78 iteration(s) Lucas sequence for Silver ratio where b = 2: First 15 elements: 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 Approximated value: 2.41421356237309504880168872420969 - reached after 44 iteration(s) Lucas sequence for Bronze ratio where b = 3: First 15 elements: 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 Approximated value: 3.30277563773199464655961063373524 - reached after 32 iteration(s) Lucas sequence for Copper ratio where b = 4: First 15 elements: 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 Approximated value: 4.23606797749978969640917366873127 - reached after 27 iteration(s) Lucas sequence for Nickel ratio where b = 5: First 15 elements: 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 Approximated value: 5.19258240356725201562535524577016 - reached after 24 iteration(s) Lucas sequence for Aluminum ratio where b = 6: First 15 elements: 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 Approximated value: 6.16227766016837933199889354443271 - reached after 22 iteration(s) Lucas sequence for Iron ratio where b = 7: First 15 elements: 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 Approximated value: 7.14005494464025913554865124576351 - reached after 20 iteration(s) Lucas sequence for Tin ratio where b = 8: First 15 elements: 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 Approximated value: 8.12310562561766054982140985597407 - reached after 19 iteration(s) Lucas sequence for Lead ratio where b = 9: First 15 elements: 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 Approximated value: 9.10977222864644365500113714088139 - reached after 18 iteration(s)
Go
package main
import (
"fmt"
"math/big"
)
var names = [10]string{"Platinum", "Golden", "Silver", "Bronze", "Copper",
"Nickel", "Aluminium", "Iron", "Tin", "Lead"}
func lucas(b int64) {
fmt.Printf("Lucas sequence for %s ratio, where b = %d:\n", names[b], b)
fmt.Print("First 15 elements: ")
var x0, x1 int64 = 1, 1
fmt.Printf("%d, %d", x0, x1)
for i := 1; i <= 13; i++ {
x2 := b*x1 + x0
fmt.Printf(", %d", x2)
x0, x1 = x1, x2
}
fmt.Println()
}
func metallic(b int64, dp int) {
x0, x1, x2, bb := big.NewInt(1), big.NewInt(1), big.NewInt(0), big.NewInt(b)
ratio := big.NewRat(1, 1)
iters := 0
prev := ratio.FloatString(dp)
for {
iters++
x2.Mul(bb, x1)
x2.Add(x2, x0)
this := ratio.SetFrac(x2, x1).FloatString(dp)
if prev == this {
plural := "s"
if iters == 1 {
plural = " "
}
fmt.Printf("Value to %d dp after %2d iteration%s: %s\n\n", dp, iters, plural, this)
return
}
prev = this
x0.Set(x1)
x1.Set(x2)
}
}
func main() {
for b := int64(0); b < 10; b++ {
lucas(b)
metallic(b, 32)
}
fmt.Println("Golden ratio, where b = 1:")
metallic(1, 256)
}
- Output:
Lucas sequence for Platinum ratio, where b = 0: First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Value to 32 dp after 1 iteration : 1.00000000000000000000000000000000 Lucas sequence for Golden ratio, where b = 1: First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 Value to 32 dp after 78 iterations: 1.61803398874989484820458683436564 Lucas sequence for Silver ratio, where b = 2: First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243 Value to 32 dp after 44 iterations: 2.41421356237309504880168872420970 Lucas sequence for Bronze ratio, where b = 3: First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564 Value to 32 dp after 34 iterations: 3.30277563773199464655961063373525 Lucas sequence for Copper ratio, where b = 4: First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141 Value to 32 dp after 28 iterations: 4.23606797749978969640917366873128 Lucas sequence for Nickel ratio, where b = 5: First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686 Value to 32 dp after 25 iterations: 5.19258240356725201562535524577016 Lucas sequence for Aluminium ratio, where b = 6: First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775 Value to 32 dp after 23 iterations: 6.16227766016837933199889354443272 Lucas sequence for Iron ratio, where b = 7: First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408 Value to 32 dp after 22 iterations: 7.14005494464025913554865124576352 Lucas sequence for Tin ratio, where b = 8: First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449 Value to 32 dp after 20 iterations: 8.12310562561766054982140985597408 Lucas sequence for Lead ratio, where b = 9: First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226 Value to 32 dp after 20 iterations: 9.10977222864644365500113714088140 Golden ratio, where b = 1: Value to 256 dp after 615 iterations: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
Groovy
class MetallicRatios {
private static List<String> names = new ArrayList<>()
static {
names.add("Platinum")
names.add("Golden")
names.add("Silver")
names.add("Bronze")
names.add("Copper")
names.add("Nickel")
names.add("Aluminum")
names.add("Iron")
names.add("Tin")
names.add("Lead")
}
private static void lucas(long b) {
printf("Lucas sequence for %s ratio, where b = %d\n", names[b], b)
print("First 15 elements: ")
long x0 = 1
long x1 = 1
printf("%d, %d", x0, x1)
for (int i = 1; i < 13; ++i) {
long x2 = b * x1 + x0
printf(", %d", x2)
x0 = x1
x1 = x2
}
println()
}
private static void metallic(long b, int dp) {
BigInteger x0 = BigInteger.ONE
BigInteger x1 = BigInteger.ONE
BigInteger x2
BigInteger bb = BigInteger.valueOf(b)
BigDecimal ratio = BigDecimal.ONE.setScale(dp)
int iters = 0
String prev = ratio.toString()
while (true) {
iters++
x2 = bb * x1 + x0
String thiz = (x2.toBigDecimal().setScale(dp) / x1.toBigDecimal().setScale(dp)).toString()
if (prev == thiz) {
String plural = "s"
if (iters == 1) {
plural = ""
}
printf("Value after %d iteration%s: %s\n\n", iters, plural, thiz)
return
}
prev = thiz
x0 = x1
x1 = x2
}
}
static void main(String[] args) {
for (int b = 0; b < 10; ++b) {
lucas(b)
metallic(b, 32)
}
println("Golden ratio, where b = 1:")
metallic(1, 256)
}
}
- Output:
Lucas sequence for Platinum ratio, where b = 0 First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Value after 2 iterations: 1 Lucas sequence for Golden ratio, where b = 1 First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 Value after 78 iterations: 1.61803398874989484820458683436564 Lucas sequence for Silver ratio, where b = 2 First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321 Value after 44 iterations: 2.41421356237309504880168872420970 Lucas sequence for Bronze ratio, where b = 3 First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601 Value after 34 iterations: 3.30277563773199464655961063373525 Lucas sequence for Copper ratio, where b = 4 First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169 Value after 28 iterations: 4.23606797749978969640917366873128 Lucas sequence for Nickel ratio, where b = 5 First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681 Value after 25 iterations: 5.19258240356725201562535524577016 Lucas sequence for Aluminum ratio, where b = 6 First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337 Value after 23 iterations: 6.16227766016837933199889354443272 Lucas sequence for Iron ratio, where b = 7 First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201 Value after 22 iterations: 7.14005494464025913554865124576352 Lucas sequence for Tin ratio, where b = 8 First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001 Value after 20 iterations: 8.12310562561766054982140985597408 Lucas sequence for Lead ratio, where b = 9 First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049 Value after 20 iterations: 9.10977222864644365500113714088140 Golden ratio, where b = 1: Value after 615 iterations: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
J
It's perhaps worth noting that we can compute the fibonacci ratio to 256 digits in ten iterations, if we use an appropriate form of iteration:
258j256":%~/+/+/ .*~^:10 x:*i.2 2
1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
However, the task implies we should use a different form of iteration, so we should probably stick to that here.
task=:{{
32 task y
:
echo 'b=',":y
echo 'seq=',":(,(1,y) X _2{.])^:15]1 1x
k=. 0
prev=.val=. ''
rat=. 1 1x
whilst.-.prev-:val do.
k=. k+1
prev=. val
rat=. }.rat,rat X 1,y
val=. (j./2 0+x)":%~/rat
end.
echo (":k),' iterations: ',val
}}
task 0
b=0
seq=1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 iterations: 1.00000000000000000000000000000000
task 1
b=1
seq=1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597
78 iterations: 1.61803398874989484820458683436564
task 2
b=2
seq=1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 275807 665857
44 iterations: 2.41421356237309504880168872420970
task 3
b=3
seq=1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 21932293 72437443
34 iterations: 3.30277563773199464655961063373525
task 4
b=4
seq=1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 701408733 2971215073
28 iterations: 4.23606797749978969640917366873128
task 5
b=5
seq=1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 11913527111 61861971241
25 iterations: 5.19258240356725201562535524577016
task 6
b=6
seq=1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 128943316987 794584521697
23 iterations: 6.16227766016837933199889354443272
task 7
b=7
seq=1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 1000716717057 7145172343807
22 iterations: 7.14005494464025913554865124576352
task 8
b=8
seq=1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 6025563297593 48946287120193
20 iterations: 8.12310562561766054982140985597408
task 9
b=9
seq=1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 29726047448083 270797521509973
20 iterations: 9.10977222864644365500113714088140
256 task 1
b=1
seq=1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597
615 iterations: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
Java
import java.math.BigDecimal;
import java.math.BigInteger;
import java.math.MathContext;
import java.util.ArrayList;
import java.util.List;
public class MetallicRatios {
private static String[] ratioDescription = new String[] {"Platinum", "Golden", "Silver", "Bronze", "Copper", "Nickel", "Aluminum", "Iron", "Tin", "Lead"};
public static void main(String[] args) {
int elements = 15;
for ( int b = 0 ; b < 10 ; b++ ) {
System.out.printf("Lucas sequence for %s ratio, where b = %d:%n", ratioDescription[b], b);
System.out.printf("First %d elements: %s%n", elements, lucasSequence(1, 1, b, elements));
int decimalPlaces = 32;
BigDecimal[] ratio = lucasSequenceRatio(1, 1, b, decimalPlaces+1);
System.out.printf("Value to %d decimal places after %s iterations : %s%n", decimalPlaces, ratio[1], ratio[0]);
System.out.printf("%n");
}
int b = 1;
int decimalPlaces = 256;
System.out.printf("%s ratio, where b = %d:%n", ratioDescription[b], b);
BigDecimal[] ratio = lucasSequenceRatio(1, 1, b, decimalPlaces+1);
System.out.printf("Value to %d decimal places after %s iterations : %s%n", decimalPlaces, ratio[1], ratio[0]);
}
private static BigDecimal[] lucasSequenceRatio(int x0, int x1, int b, int digits) {
BigDecimal x0Bi = BigDecimal.valueOf(x0);
BigDecimal x1Bi = BigDecimal.valueOf(x1);
BigDecimal bBi = BigDecimal.valueOf(b);
MathContext mc = new MathContext(digits);
BigDecimal fractionPrior = x1Bi.divide(x0Bi, mc);
int iterations = 0;
while ( true ) {
iterations++;
BigDecimal x = bBi.multiply(x1Bi).add(x0Bi);
BigDecimal fractionCurrent = x.divide(x1Bi, mc);
if ( fractionCurrent.compareTo(fractionPrior) == 0 ) {
break;
}
x0Bi = x1Bi;
x1Bi = x;
fractionPrior = fractionCurrent;
}
return new BigDecimal[] {fractionPrior, BigDecimal.valueOf(iterations)};
}
private static List<BigInteger> lucasSequence(int x0, int x1, int b, int n) {
List<BigInteger> list = new ArrayList<>();
BigInteger x0Bi = BigInteger.valueOf(x0);
BigInteger x1Bi = BigInteger.valueOf(x1);
BigInteger bBi = BigInteger.valueOf(b);
if ( n > 0 ) {
list.add(x0Bi);
}
if ( n > 1 ) {
list.add(x1Bi);
}
while ( n > 2 ) {
BigInteger x = bBi.multiply(x1Bi).add(x0Bi);
list.add(x);
n--;
x0Bi = x1Bi;
x1Bi = x;
}
return list;
}
}
- Output:
Lucas sequence for Platinum ratio, where b = 0: First 15 elements: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Value to 32 decimal places after 1 iterations : 1 Lucas sequence for Golden ratio, where b = 1: First 15 elements: [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610] Value to 32 decimal places after 78 iterations : 1.61803398874989484820458683436564 Lucas sequence for Silver ratio, where b = 2: First 15 elements: [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243] Value to 32 decimal places after 44 iterations : 2.41421356237309504880168872420970 Lucas sequence for Bronze ratio, where b = 3: First 15 elements: [1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564] Value to 32 decimal places after 34 iterations : 3.30277563773199464655961063373525 Lucas sequence for Copper ratio, where b = 4: First 15 elements: [1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141] Value to 32 decimal places after 28 iterations : 4.23606797749978969640917366873128 Lucas sequence for Nickel ratio, where b = 5: First 15 elements: [1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686] Value to 32 decimal places after 25 iterations : 5.19258240356725201562535524577016 Lucas sequence for Aluminum ratio, where b = 6: First 15 elements: [1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775] Value to 32 decimal places after 23 iterations : 6.16227766016837933199889354443272 Lucas sequence for Iron ratio, where b = 7: First 15 elements: [1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408] Value to 32 decimal places after 22 iterations : 7.14005494464025913554865124576352 Lucas sequence for Tin ratio, where b = 8: First 15 elements: [1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449] Value to 32 decimal places after 20 iterations : 8.12310562561766054982140985597408 Lucas sequence for Lead ratio, where b = 9: First 15 elements: [1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226] Value to 32 decimal places after 20 iterations : 9.10977222864644365500113714088140 Golden ratio, where b = 1: Value to 256 decimal places after 615 iterations : 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
Julia
using Formatting
import Base.iterate, Base.IteratorSize, Base.IteratorEltype, Base.Iterators.take
const metallicnames = ["Platinum", "Golden", "Silver", "Bronze", "Copper", "Nickel",
"Aluminium", "Iron", "Tin", "Lead"]
struct Lucas b::Int end
Base.IteratorSize(s::Lucas) = Base.IsInfinite()
Base.IteratorEltype(s::Lucas) = BigInt
Base.iterate(s::Lucas, (x1, x2) = (big"1", big"1")) = (t = x2 * s.b + x1; (x1, (x2, t)))
printlucas(b, len=15) = (for i in take(Lucas(b), len) print(i, ", ") end; println("..."))
function lucasratios(b, len)
iter = BigFloat.(collect(take(Lucas(b), len + 1)))
return map(i -> iter[i + 1] / iter[i], 1:length(iter)-1)
end
function metallic(b, dplaces=32)
setprecision(dplaces * 5)
ratios, err = lucasratios(b, dplaces * 50), BigFloat(10)^(-dplaces)
errors = map(i -> abs(ratios[i + 1] - ratios[i]), 1:length(ratios)-1)
iternum = findfirst(x -> x < err, errors)
println("After $(iternum + 1) iterations, the value of ",
format(ratios[iternum + 1], precision=dplaces),
" is stable to $dplaces decimal places.\n")
end
for (b, name) in enumerate(metallicnames)
println("The first 15 elements of the Lucas sequence named ",
metallicnames[b], " and b of $(b - 1) are:")
printlucas(b - 1)
metallic(b - 1)
end
println("Golden ratio to 256 decimal places:")
metallic(1, 256)
- Output:
The first 15 elements of the Lucas sequence named Platinum and b of 0 are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... After 2 iterations, the value of 1.00000000000000000000000000000000 is stable to 32 decimal places. The first 15 elements of the Lucas sequence named Golden and b of 1 are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ... After 79 iterations, the value of 1.61803398874989484820458683436564 is stable to 32 decimal places. The first 15 elements of the Lucas sequence named Silver and b of 2 are: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243, ... After 45 iterations, the value of 2.41421356237309504880168872420970 is stable to 32 decimal places. The first 15 elements of the Lucas sequence named Bronze and b of 3 are: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564, ... After 34 iterations, the value of 3.30277563773199464655961063373525 is stable to 32 decimal places. The first 15 elements of the Lucas sequence named Copper and b of 4 are: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141, ... After 29 iterations, the value of 4.23606797749978969640917366873128 is stable to 32 decimal places. The first 15 elements of the Lucas sequence named Nickel and b of 5 are: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686, ... After 26 iterations, the value of 5.19258240356725201562535524577016 is stable to 32 decimal places. The first 15 elements of the Lucas sequence named Aluminium and b of 6 are: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775, ... After 24 iterations, the value of 6.16227766016837933199889354443272 is stable to 32 decimal places. The first 15 elements of the Lucas sequence named Iron and b of 7 are: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408, ... After 22 iterations, the value of 7.14005494464025913554865124576352 is stable to 32 decimal places. The first 15 elements of the Lucas sequence named Tin and b of 8 are: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449, ... After 21 iterations, the value of 8.12310562561766054982140985597408 is stable to 32 decimal places. The first 15 elements of the Lucas sequence named Lead and b of 9 are: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226, ... After 20 iterations, the value of 9.10977222864644365500113714088140 is stable to 32 decimal places. Golden ratio to 256 decimal places: After 615 iterations, the value of 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144 is stable to 256 decimal places.
Kotlin
import java.math.BigDecimal
import java.math.BigInteger
val names = listOf("Platinum", "Golden", "Silver", "Bronze", "Copper", "Nickel", "Aluminium", "Iron", "Tin", "Lead")
fun lucas(b: Long) {
println("Lucas sequence for ${names[b.toInt()]} ratio, where b = $b:")
print("First 15 elements: ")
var x0 = 1L
var x1 = 1L
print("$x0, $x1")
for (i in 1..13) {
val x2 = b * x1 + x0
print(", $x2")
x0 = x1
x1 = x2
}
println()
}
fun metallic(b: Long, dp:Int) {
var x0 = BigInteger.ONE
var x1 = BigInteger.ONE
var x2: BigInteger
val bb = BigInteger.valueOf(b)
val ratio = BigDecimal.ONE.setScale(dp)
var iters = 0
var prev = ratio.toString()
while (true) {
iters++
x2 = bb * x1 + x0
val thiz = (x2.toBigDecimal(dp) / x1.toBigDecimal(dp)).toString()
if (prev == thiz) {
var plural = "s"
if (iters == 1) {
plural = ""
}
println("Value after $iters iteration$plural: $thiz\n")
return
}
prev = thiz
x0 = x1
x1 = x2
}
}
fun main() {
for (b in 0L until 10L) {
lucas(b)
metallic(b, 32)
}
println("Golden ration, where b = 1:")
metallic(1, 256)
}
- Output:
Lucas sequence for Platinum ratio, where b = 0: First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Value after 1 iteration: 1.00000000000000000000000000000000 Lucas sequence for Golden ratio, where b = 1: First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 Value after 78 iterations: 1.61803398874989484820458683436564 Lucas sequence for Silver ratio, where b = 2: First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243 Value after 44 iterations: 2.41421356237309504880168872420970 Lucas sequence for Bronze ratio, where b = 3: First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564 Value after 34 iterations: 3.30277563773199464655961063373525 Lucas sequence for Copper ratio, where b = 4: First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141 Value after 28 iterations: 4.23606797749978969640917366873128 Lucas sequence for Nickel ratio, where b = 5: First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686 Value after 25 iterations: 5.19258240356725201562535524577016 Lucas sequence for Aluminium ratio, where b = 6: First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775 Value after 23 iterations: 6.16227766016837933199889354443272 Lucas sequence for Iron ratio, where b = 7: First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408 Value after 22 iterations: 7.14005494464025913554865124576352 Lucas sequence for Tin ratio, where b = 8: First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449 Value after 20 iterations: 8.12310562561766054982140985597408 Lucas sequence for Lead ratio, where b = 9: First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226 Value after 20 iterations: 9.10977222864644365500113714088140 Golden ration, where b = 1: Value after 615 iterations: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
Mathematica/Wolfram Language
ClearAll[FindMetallicRatio]
FindMetallicRatio[b_, digits_] :=
Module[{n, m, data, acc, old, done = False},
{n, m} = {1, 1};
old = -100;
data = {};
While[done == False,
{n, m} = {m, b m + n};
AppendTo[data, {m, m/n}];
If[Length[data] > 15,
If[-N[Log10[Abs[data[[-1, 2]] - data[[-2, 2]]]]] > digits,
done = True
]
]
];
acc = -N[Log10[Abs[data[[-1, 2]] - data[[-2, 2]]]]];
<|"sequence" -> Join[{1, 1}, data[[All, 1]]],
"ratio" -> data[[All, 2]], "acc" -> acc,
"steps" -> Length[data]|>
]
Do[
out = FindMetallicRatio[b, 32];
Print["b=", b];
Print["b=", b, " first 15=", Take[out["sequence"], 15]];
Print["b=", b, " ratio=", N[Last[out["ratio"]], {\[Infinity], 33}]];
Print["b=", b, " Number of steps=", out["steps"]];
,
{b, 0, 9}
]
out = FindMetallicRatio[1, 256];
out["steps"]
N[out["ratio"][[-1]], 256]
- Output:
b=0 b=0 first 15={1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} b=0 ratio=1.00000000000000000000000000000000 b=0 Number of steps=16 b=1 b=1 first 15={1,1,2,3,5,8,13,21,34,55,89,144,233,377,610} b=1 ratio=1.61803398874989484820458683436564 b=1 Number of steps=78 b=2 b=2 first 15={1,1,3,7,17,41,99,239,577,1393,3363,8119,19601,47321,114243} b=2 ratio=2.41421356237309504880168872420970 b=2 Number of steps=44 b=3 b=3 first 15={1,1,4,13,43,142,469,1549,5116,16897,55807,184318,608761,2010601,6640564} b=3 ratio=3.30277563773199464655961063373525 b=3 Number of steps=33 b=4 b=4 first 15={1,1,5,21,89,377,1597,6765,28657,121393,514229,2178309,9227465,39088169,165580141} b=4 ratio=4.236067977499789696409173668731276 b=4 Number of steps=28 b=5 b=5 first 15={1,1,6,31,161,836,4341,22541,117046,607771,3155901,16387276,85092281,441848681,2294335686} b=5 ratio=5.19258240356725201562535524577016 b=5 Number of steps=25 b=6 b=6 first 15={1,1,7,43,265,1633,10063,62011,382129,2354785,14510839,89419819,551029753,3395598337,20924619775} b=6 ratio=6.162277660168379331998893544432719 b=6 Number of steps=23 b=7 b=7 first 15={1,1,8,57,407,2906,20749,148149,1057792,7552693,53926643,385039194,2749201001,19629446201,140155324408} b=7 ratio=7.14005494464025913554865124576352 b=7 Number of steps=21 b=8 b=8 first 15={1,1,9,73,593,4817,39129,317849,2581921,20973217,170367657,1383914473,11241683441,91317382001,741780739449} b=8 ratio=8.123105625617660549821409855974077 b=8 Number of steps=20 b=9 b=9 first 15={1,1,10,91,829,7552,68797,626725,5709322,52010623,473804929,4316254984,39320099785,358197153049,3263094477226} b=9 ratio=9.10977222864644365500113714088140 b=9 Number of steps=19 614 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614
Nim
import strformat
import bignum
type Metal {.pure.} = enum platinum, golden, silver, bronze, copper, nickel, aluminium, iron, tin, lead
iterator sequence(b: int): Int =
## Yield the successive terms if a “Lucas” sequence.
## The first two terms are ignored.
var x, y = newInt(1)
while true:
x += b * y
swap x, y
yield y
template plural(n: int): string =
if n >= 2: "s" else: ""
proc computeRatio(b: Natural; digits: Positive) =
## Compute the ratio for the given "n" with the required number of digits.
let M = 10^culong(digits)
var niter = 0 # Number of iterations.
var prevN = newInt(1) # Previous value of "n".
var ratio = M # Current value of ratio.
for n in sequence(b):
inc niter
let nextRatio = n * M div prevN
if nextRatio == ratio: break
prevN = n.clone
ratio = nextRatio
var str = $ratio
str.insert(".", 1)
echo &"Value to {digits} decimal places after {niter} iteration{plural(niter)}: ", str
when isMainModule:
for b in 0..9:
echo &"“Lucas” sequence for {Metal(b)} ratio where b = {b}:"
stdout.write "First 15 elements: 1 1"
var count = 2
for n in sequence(b):
stdout.write ' ', n
inc count
if count == 15: break
echo ""
computeRatio(b, 32)
echo ""
echo "Golden ratio where b = 1:"
computeRatio(1, 256)
- Output:
“Lucas” sequence for platinum ratio where b = 0: First 15 elements: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Value to 32 decimal places after 1 iteration: 1.00000000000000000000000000000000 “Lucas” sequence for golden ratio where b = 1: First 15 elements: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 Value to 32 decimal places after 79 iterations: 1.61803398874989484820458683436563 “Lucas” sequence for silver ratio where b = 2: First 15 elements: 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 Value to 32 decimal places after 45 iterations: 2.41421356237309504880168872420969 “Lucas” sequence for bronze ratio where b = 3: First 15 elements: 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 Value to 32 decimal places after 33 iterations: 3.30277563773199464655961063373524 “Lucas” sequence for copper ratio where b = 4: First 15 elements: 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 Value to 32 decimal places after 28 iterations: 4.23606797749978969640917366873127 “Lucas” sequence for nickel ratio where b = 5: First 15 elements: 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 Value to 32 decimal places after 25 iterations: 5.19258240356725201562535524577016 “Lucas” sequence for aluminium ratio where b = 6: First 15 elements: 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 Value to 32 decimal places after 23 iterations: 6.16227766016837933199889354443271 “Lucas” sequence for iron ratio where b = 7: First 15 elements: 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 Value to 32 decimal places after 21 iterations: 7.14005494464025913554865124576351 “Lucas” sequence for tin ratio where b = 8: First 15 elements: 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 Value to 32 decimal places after 20 iterations: 8.12310562561766054982140985597407 “Lucas” sequence for lead ratio where b = 9: First 15 elements: 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 Value to 32 decimal places after 19 iterations: 9.10977222864644365500113714088139 Golden ratio where b = 1: Value to 256 decimal places after 614 iterations: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
Perl
use strict;
use warnings;
use feature qw(say state);
use Math::AnyNum qw<:overload as_dec>;
sub gen_lucas {
my $b = shift;
my $i = 0;
return sub {
state @seq = (state $v1 = 1, state $v2 = 1);
($v2, $v1) = ($v1, $v2 + $b*$v1) and push(@seq, $v1) unless defined $seq[$i+1];
return $seq[$i++];
}
}
sub metallic {
my $lucas = shift;
my $places = shift || 32;
my $n = my $last = 0;
my @seq = $lucas->();
while (1) {
push @seq, $lucas->();
my $this = as_dec( $seq[-1]/$seq[-2], $places+1 );
last if $this eq $last;
$last = $this;
$n++;
}
$last, $n
}
my @name = <Platinum Golden Silver Bronze Copper Nickel Aluminum Iron Tin Lead>;
for my $b (0..$#name) {
my $lucas = gen_lucas($b);
printf "\n'Lucas' sequence for $name[$b] ratio, where b = $b:\nFirst 15 elements: " . join ', ', map { $lucas->() } 1..15;
printf "Approximated value %s reached after %d iterations\n", metallic(gen_lucas($b));
}
printf "\nGolden ratio to 256 decimal places %s reached after %d iterations", metallic(gen_lucas(1),256);
- Output:
'Lucas' sequence for Platinum ratio, where b = 0: First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Approximated value 1 reached after 1 iterations 'Lucas' sequence for Golden ratio, where b = 1: First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 Approximated value 1.61803398874989484820458683436564 reached after 78 iterations 'Lucas' sequence for Silver ratio, where b = 2: First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243 Approximated value 2.4142135623730950488016887242097 reached after 44 iterations 'Lucas' sequence for Bronze ratio, where b = 3: First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564 Approximated value 3.30277563773199464655961063373525 reached after 34 iterations 'Lucas' sequence for Copper ratio, where b = 4: First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141 Approximated value 4.23606797749978969640917366873128 reached after 28 iterations 'Lucas' sequence for Nickel ratio, where b = 5: First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686 Approximated value 5.19258240356725201562535524577016 reached after 25 iterations 'Lucas' sequence for Aluminum ratio, where b = 6: First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775 Approximated value 6.16227766016837933199889354443272 reached after 23 iterations 'Lucas' sequence for Iron ratio, where b = 7: First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408 Approximated value 7.14005494464025913554865124576352 reached after 22 iterations 'Lucas' sequence for Tin ratio, where b = 8: First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449 Approximated value 8.12310562561766054982140985597408 reached after 20 iterations 'Lucas' sequence for Lead ratio, where b = 9: First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226 Approximated value 9.1097722286464436550011371408814 reached after 20 iterations Golden ratio to 256 decimal places 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144 reached after 615 iterations
Phix
with javascript_semantics include mpfr.e constant names = {"Platinum", "Golden", "Silver", "Bronze", "Copper", "Nickel", "Aluminium", "Iron", "Tin", "Lead"} procedure lucas(integer b) printf(1,"Lucas sequence for %s ratio, where b = %d:\n", {names[b+1], b}) atom x0 = 1, x1 = 1, x2 printf(1,"First 15 elements: %d, %d", {x0, x1}) for i=1 to 13 do x2 = b*x1 + x0 printf(1,", %d", x2) x0 = x1 x1 = x2 end for printf(1,"\n") end procedure procedure metallic(integer b, dp) mpz {x0, x1, x2, bb} = mpz_inits(4,{1,1,0,b}) mpfr ratio = mpfr_init(1,-(dp+2)) integer iterations = 0 string prev = mpfr_get_fixed(ratio,dp) while true do iterations += 1 mpz_mul(x2,bb,x1) mpz_add(x2,x2,x0) mpfr_set_z(ratio,x2) mpfr_div_z(ratio,ratio,x1) string curr = mpfr_get_fixed(ratio,dp) if prev == curr then string plural = iff(iterations=1?"":"s") curr = shorten(curr) printf(1,"Value to %d dp after %2d iteration%s: %s\n\n", {dp, iterations, plural, curr}) exit end if prev = curr mpz_set(x0,x1) mpz_set(x1,x2) end while end procedure procedure main() for b=0 to 9 do lucas(b) metallic(b, 32) end for printf(1,"Golden ratio, where b = 1:\n") metallic(1, 256) end procedure main()
- Output:
The final 258 digits includes the "1.", and dp+2 was needed on ratio to exactly match the Go (etc) output.
Lucas sequence for Platinum ratio, where b = 0: First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Value to 32 dp after 1 iteration: 1.00000000000000000000000000000000 Lucas sequence for Golden ratio, where b = 1: First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 Value to 32 dp after 78 iterations: 1.61803398874989484820458683436564 Lucas sequence for Silver ratio, where b = 2: First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243 Value to 32 dp after 44 iterations: 2.41421356237309504880168872420970 Lucas sequence for Bronze ratio, where b = 3: First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564 Value to 32 dp after 34 iterations: 3.30277563773199464655961063373525 Lucas sequence for Copper ratio, where b = 4: First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141 Value to 32 dp after 28 iterations: 4.23606797749978969640917366873128 Lucas sequence for Nickel ratio, where b = 5: First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686 Value to 32 dp after 25 iterations: 5.19258240356725201562535524577016 Lucas sequence for Aluminium ratio, where b = 6: First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775 Value to 32 dp after 23 iterations: 6.16227766016837933199889354443272 Lucas sequence for Iron ratio, where b = 7: First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408 Value to 32 dp after 22 iterations: 7.14005494464025913554865124576352 Lucas sequence for Tin ratio, where b = 8: First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449 Value to 32 dp after 20 iterations: 8.12310562561766054982140985597408 Lucas sequence for Lead ratio, where b = 9: First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226 Value to 32 dp after 20 iterations: 9.10977222864644365500113714088140 Golden ratio, where b = 1: Value to 256 dp after 615 iterations: 1.61803398874989484...2695486262963136144 (258 digits)
Python
from itertools import count, islice
from _pydecimal import getcontext, Decimal
def metallic_ratio(b):
m, n = 1, 1
while True:
yield m, n
m, n = m*b + n, m
def stable(b, prec):
def to_decimal(b):
for m,n in metallic_ratio(b):
yield Decimal(m)/Decimal(n)
getcontext().prec = prec
last = 0
for i,x in zip(count(), to_decimal(b)):
if x == last:
print(f'after {i} iterations:\n\t{x}')
break
last = x
for b in range(4):
coefs = [n for _,n in islice(metallic_ratio(b), 15)]
print(f'\nb = {b}: {coefs}')
stable(b, 32)
print(f'\nb = 1 with 256 digits:')
stable(1, 256)
- Output:
b = 0: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] after 1 iterations: 1 b = 1: [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610] after 77 iterations: 1.6180339887498948482045868343656 b = 2: [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243] after 43 iterations: 2.4142135623730950488016887242097 b = 3: [1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564] after 33 iterations: 3.3027756377319946465596106337352 b = 1 with 256 digits: after 613 iterations: 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614
Quackery
[ $ "bigrat.qky" loadfile ] now!
[ [ table
$ "Platinum" $ "Golden"
$ "Silver" $ "Bronze"
$ "Copper" $ "Nickel"
$ "Aluminium" $ "Iron"
$ "Tin" $ "Lead" ]
do echo$ ] is echoname ( n --> )
[ temp put
' [ 1 1 ]
13 times
[ dup -1 peek
temp share *
over -2 peek +
join ]
temp release ] is task1 ( n --> [ )
[ temp put
' [ 0 1 1 ]
[ dup -3 split nip
do unrot tuck swap
32 approx= if done
dup -1 peek
temp share *
over -2 peek +
join
again ]
temp release
dup size 3 - swap
-3 split nip
-1 split drop
do swap rot ] is task2 ( n --> n/d n )
10 times
[ i^ echoname cr
i^ task1
witheach [ echo sp ] cr
i^ task2
echo sp
32 point$ echo$
cr cr ]
- Output:
Platinum 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Golden 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 78 1.61803398874989484820458683436564 Silver 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 44 2.4142135623730950488016887242097 Bronze 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 33 3.30277563773199464655961063373524 Copper 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 28 4.23606797749978969640917366873128 Nickel 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 25 5.19258240356725201562535524577016 Aluminium 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 23 6.16227766016837933199889354443272 Iron 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 21 7.14005494464025913554865124576351 Tin 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 20 8.12310562561766054982140985597408 Lead 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 19 9.10977222864644365500113714088139
Raku
(formerly Perl 6)
Note: Arrays, Lists and Sequences are zero indexed in Raku.
use Rat::Precise;
use Lingua::EN::Numbers;
sub lucas ($b) { 1, 1, * + $b * * … * }
sub metallic ($seq, $places = 32) {
my $n = 0;
my $last = 0;
loop {
my $approx = FatRat.new($seq[$n + 1], $seq[$n]);
my $this = $approx.precise($places, :z);
last if $this eq $last;
$last = $this;
$n++;
}
$last, $n
}
sub display ($value, $n) {
"Approximated value:", $value, "Reached after {$n} iterations: " ~
"{ordinal-digit $n}/{ordinal-digit $n - 1} element."
}
for <Platinum Golden Silver Bronze Copper Nickel Aluminum Iron Tin Lead>.kv
-> \b, $name {
my $lucas = lucas b;
print "\nLucas sequence for $name ratio; where b = {b}:\nFirst 15 elements: ";
say join ', ', $lucas[^15];
say join ' ', display |metallic($lucas);
}
# Stretch goal
say join "\n", "\nGolden ratio to 256 decimal places:", display |metallic lucas(1), 256;
- Output:
Lucas sequence for Platinum ratio; where b = 0: First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Approximated value: 1.00000000000000000000000000000000 Reached after 1 iterations: 1st/0th element. Lucas sequence for Golden ratio; where b = 1: First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 Approximated value: 1.61803398874989484820458683436564 Reached after 78 iterations: 78th/77th element. Lucas sequence for Silver ratio; where b = 2: First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243 Approximated value: 2.41421356237309504880168872420970 Reached after 44 iterations: 44th/43rd element. Lucas sequence for Bronze ratio; where b = 3: First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564 Approximated value: 3.30277563773199464655961063373525 Reached after 34 iterations: 34th/33rd element. Lucas sequence for Copper ratio; where b = 4: First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141 Approximated value: 4.23606797749978969640917366873128 Reached after 28 iterations: 28th/27th element. Lucas sequence for Nickel ratio; where b = 5: First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686 Approximated value: 5.19258240356725201562535524577016 Reached after 25 iterations: 25th/24th element. Lucas sequence for Aluminum ratio; where b = 6: First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775 Approximated value: 6.16227766016837933199889354443272 Reached after 23 iterations: 23rd/22nd element. Lucas sequence for Iron ratio; where b = 7: First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408 Approximated value: 7.14005494464025913554865124576352 Reached after 22 iterations: 22nd/21st element. Lucas sequence for Tin ratio; where b = 8: First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449 Approximated value: 8.12310562561766054982140985597408 Reached after 20 iterations: 20th/19th element. Lucas sequence for Lead ratio; where b = 9: First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226 Approximated value: 9.10977222864644365500113714088140 Reached after 20 iterations: 20th/19th element. Golden ratio to 256 decimal places: Approximated value: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144 Reached after 615 iterations: 615th/614th element.
REXX
For this task, the elements of the Lucas sequence are zero based.
/*REXX pgm computes the 1st N elements of the Lucas sequence for Metallic ratios 0──►9. */
parse arg n bLO bHI digs . /*obtain optional arguments from the CL*/
if n=='' | n=="," then n= 15 /*Not specified? Then use the default.*/
if bLO=='' | bLO=="," then bLO= 0 /* " " " " " " */
if bHI=='' | bHI=="," then bHI= 9 /* " " " " " " */
if digs=='' | digs=="," then digs= 32 /* " " " " " " */
numeric digits digs + length(.) /*specify number of decimal digs to use*/
metals= 'platinum golden silver bronze copper nickel aluminum iron tin lead'
@decDigs= ' decimal digits past the decimal point:' /*a literal used in SAY.*/
!.= /*the default name for a metallic ratio*/
do k=0 to 9; !.k= word(metals, k+1) /*assign the (ten) metallic ratio names*/
end /*k*/
do m=bLO to bHI; @.= 1; $= 1 1 /*compute the sequence numbers & ratios*/
r=. /*the ratio (so far). */
do #=2 until r=old; old= r /*compute sequence numbers & the ratio.*/
#_1= #-1; #_2= #-2 /*use variables for previous numbers. */
@.#= m * @.#_1 + @.#_2 /*calculate a number i the sequence. */
if #<n then $= $ @.# /*build a sequence list of N numbers.*/
r= @.# / @.#_1 /*calculate ratio of the last 2 numbers*/
end /*#*/
if words($)<n then $= subword($ copies('1 ', n), 1, n) /*extend list if too short*/
L= max(108, length($) ) /*ensure width of title. */
say center(' Lucas sequence for the' !.m "ratio, where B is " m' ', L, "═")
if n>0 then do; say 'the first ' n " elements are:"; say $
end /*if N is positive, then show N nums.*/
@approx= 'approximate' /*literal (1 word) that is used for SAY*/
r= format(r,,digs) /*limit decimal digits for R to digs.*/
if datatype(r, 'W') then do; r= r/1; @approx= "exact"; end
say 'the' @approx "value reached after" #-1 " iterations with " digs @DecDigs
say r; say /*display the ration plus a blank line.*/
end /*m*/ /*stick a fork in it, we're all done. */
- output when using the default inputs:
Shown at three-quarter size.)
═════════════════════════ Lucas sequence for the platinum ratio, where B is 0 ══════════════════════════ the first 15 elements are: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 the exact value reached after 2 iterations with 32 decimal digits past the decimal point: 1 ══════════════════════════ Lucas sequence for the golden ratio, where B is 1 ═══════════════════════════ the first 15 elements are: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 the approximate value reached after 78 iterations with 32 decimal digits past the decimal point: 1.61803398874989484820458683436564 ══════════════════════════ Lucas sequence for the silver ratio, where B is 2 ═══════════════════════════ the first 15 elements are: 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 the approximate value reached after 44 iterations with 32 decimal digits past the decimal point: 2.41421356237309504880168872420970 ══════════════════════════ Lucas sequence for the bronze ratio, where B is 3 ═══════════════════════════ the first 15 elements are: 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 the approximate value reached after 34 iterations with 32 decimal digits past the decimal point: 3.30277563773199464655961063373525 ══════════════════════════ Lucas sequence for the copper ratio, where B is 4 ═══════════════════════════ the first 15 elements are: 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 the approximate value reached after 28 iterations with 32 decimal digits past the decimal point: 4.23606797749978969640917366873128 ══════════════════════════ Lucas sequence for the nickel ratio, where B is 5 ═══════════════════════════ the first 15 elements are: 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 the approximate value reached after 25 iterations with 32 decimal digits past the decimal point: 5.19258240356725201562535524577016 ═════════════════════════ Lucas sequence for the aluminum ratio, where B is 6 ══════════════════════════ the first 15 elements are: 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 the approximate value reached after 23 iterations with 32 decimal digits past the decimal point: 6.16227766016837933199889354443272 ═══════════════════════════ Lucas sequence for the iron ratio, where B is 7 ════════════════════════════ the first 15 elements are: 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 the approximate value reached after 22 iterations with 32 decimal digits past the decimal point: 7.14005494464025913554865124576352 ════════════════════════════ Lucas sequence for the tin ratio, where B is 8 ════════════════════════════ the first 15 elements are: 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 the approximate value reached after 20 iterations with 32 decimal digits past the decimal point: 8.12310562561766054982140985597408 ═══════════════════════════ Lucas sequence for the lead ratio, where B is 9 ════════════════════════════ the first 15 elements are: 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 the approximate value reached after 20 iterations with 32 decimal digits past the decimal point: 9.10977222864644365500113714088140
- output when using the inputs of: 0 1 1 256
(Shown at three-quarter size.)
══════════════════════════ Lucas sequence for the golden ratio, where B is 1 ═══════════════════════════ the approximate value reached after 615 iterations with 256 decimal digits past the decimal point: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
Ruby
require('bigdecimal')
require('bigdecimal/util')
# An iterator over the Lucas Sequence for 'b'.
# (The special case of: x(n) = b * x(n-1) + x(n-2).)
def lucas(b)
Enumerator.new do |yielder|
xn2 = 1 ; yielder.yield(xn2)
xn1 = 1 ; yielder.yield(xn1)
loop { xn2, xn1 = xn1, b * xn1 + xn2 ; yielder.yield(xn1) }
end
end
# Compute the Metallic Ratio to 'precision' from the Lucas Sequence for 'b'.
# (Uses the lucas(b) iterator, above.)
# The metallic ratio is approximated by x(n) / x(n-1).
# Returns a struct of the approximate metallic ratio (.ratio) and the
# number of terms required to achieve the given precision (.terms).
def metallic_ratio(b, precision)
xn2 = xn1 = prev = this = 0
lucas(b).each.with_index do |xn, inx|
case inx
when 0
xn2 = BigDecimal(xn)
when 1
xn1 = BigDecimal(xn)
prev = xn1.div(xn2, 2 * precision).round(precision)
else
xn2, xn1 = xn1, BigDecimal(xn)
this = xn1.div(xn2, 2 * precision).round(precision)
return Struct.new(:ratio, :terms).new(prev, inx - 1) if prev == this
prev = this
end
end
end
NAMES = [ 'Platinum', 'Golden', 'Silver', 'Bronze', 'Copper',
'Nickel', 'Aluminum', 'Iron', 'Tin', 'Lead' ]
puts
puts('Lucas Sequences...')
puts('%1s %s' % ['b', 'sequence'])
(0..9).each do |b|
puts('%1d %s' % [b, lucas(b).first(15)])
end
puts
puts('Metallic Ratios to 32 places...')
puts('%-9s %1s %3s %s' % ['name', 'b', 'n', 'ratio'])
(0..9).each do |b|
rn = metallic_ratio(b, 32)
puts('%-9s %1d %3d %s' % [NAMES[b], b, rn.terms, rn.ratio.to_s('F')])
end
puts
puts('Golden Ratio to 256 places...')
puts('%-9s %1s %3s %s' % ['name', 'b', 'n', 'ratio'])
gold_rn = metallic_ratio(1, 256)
puts('%-9s %1d %3d %s' % [NAMES[1], 1, gold_rn.terms, gold_rn.ratio.to_s('F')])
- Output:
Lucas Sequences... b sequence 0 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 1 [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610] 2 [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243] 3 [1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564] 4 [1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141] 5 [1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686] 6 [1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775] 7 [1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408] 8 [1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449] 9 [1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226] Metallic Ratios to 32 places... name b n ratio Platinum 0 1 1.0 Golden 1 78 1.61803398874989484820458683436564 Silver 2 44 2.4142135623730950488016887242097 Bronze 3 34 3.30277563773199464655961063373525 Copper 4 28 4.23606797749978969640917366873128 Nickel 5 25 5.19258240356725201562535524577016 Aluminum 6 23 6.16227766016837933199889354443272 Iron 7 22 7.14005494464025913554865124576352 Tin 8 20 8.12310562561766054982140985597408 Lead 9 20 9.1097722286464436550011371408814 Golden Ratio to 256 places... name b n ratio Golden 1 615 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
Sidef
func seqRatio(f, places = 32) {
1..Inf -> reduce {|t,n|
var r = (f(n+1)/f(n)).round(-places)
return(n, r.as_dec(places + r.abs.int.len)) if (r == t)
r
}
}
for k,v in (%w(Platinum Golden Silver Bronze Copper Nickel Aluminum Iron Tin Lead).kv) {
next if (k == 0) # undefined ratio
say "Lucas sequence U_n(#{k},-1) for #{v} ratio"
var f = {|n| lucasu(k, -1, n) }
say ("First 15 elements: ", 15.of(f).join(', '))
var (n, r) = seqRatio(f)
say "Approximated value: #{r} reached after #{n} iterations"
say ''
}
with (seqRatio({|n| fib(n) }, 256)) {|n,v|
say "Golden ratio to 256 decimal places:"
say "Approximated value: #{v}"
say "Reached after #{n} iterations"
}
- Output:
Lucas sequence U_n(1,-1) for Golden ratio First 15 elements: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 Approximated value: 1.61803398874989484820458683436564 reached after 79 iterations Lucas sequence U_n(2,-1) for Silver ratio First 15 elements: 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782 Approximated value: 2.4142135623730950488016887242097 reached after 45 iterations Lucas sequence U_n(3,-1) for Bronze ratio First 15 elements: 0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243 Approximated value: 3.30277563773199464655961063373525 reached after 33 iterations Lucas sequence U_n(4,-1) for Copper ratio First 15 elements: 0, 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, 1762289, 7465176, 31622993, 133957148 Approximated value: 4.23606797749978969640917366873128 reached after 28 iterations Lucas sequence U_n(5,-1) for Nickel ratio First 15 elements: 0, 1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, 13741001, 71351280, 370497401, 1923838285 Approximated value: 5.19258240356725201562535524577016 reached after 26 iterations Lucas sequence U_n(6,-1) for Aluminum ratio First 15 elements: 0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, 76934989, 474094764, 2921503573, 18003116202 Approximated value: 6.16227766016837933199889354443272 reached after 23 iterations Lucas sequence U_n(7,-1) for Iron ratio First 15 elements: 0, 1, 7, 50, 357, 2549, 18200, 129949, 927843, 6624850, 47301793, 337737401, 2411463600, 17217982601, 122937341807 Approximated value: 7.14005494464025913554865124576352 reached after 21 iterations Lucas sequence U_n(8,-1) for Tin ratio First 15 elements: 0, 1, 8, 65, 528, 4289, 34840, 283009, 2298912, 18674305, 151693352, 1232221121, 10009462320, 81307919681, 660472819768 Approximated value: 8.12310562561766054982140985597408 reached after 20 iterations Lucas sequence U_n(9,-1) for Lead ratio First 15 elements: 0, 1, 9, 82, 747, 6805, 61992, 564733, 5144589, 46866034, 426938895, 3889316089, 35430783696, 322766369353, 2940328107873 Approximated value: 9.1097722286464436550011371408814 reached after 19 iterations Golden ratio to 256 decimal places: Approximated value: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144 Reached after 616 iterations
Visual Basic .NET
Imports BI = System.Numerics.BigInteger
Module Module1
Function IntSqRoot(v As BI, res As BI) As BI
REM res is the initial guess
Dim term As BI = 0
Dim d As BI = 0
Dim dl As BI = 1
While dl <> d
term = v / res
res = (res + term) >> 1
dl = d
d = term - res
End While
Return term
End Function
Function DoOne(b As Integer, digs As Integer) As String
REM calculates result via square root, not iterations
Dim s = b * b + 4
digs += 1
Dim g As BI = Math.Sqrt(s * Math.Pow(10, digs))
Dim bs = IntSqRoot(s * BI.Parse("1" + New String("0", digs << 1)), g)
bs += b * BI.Parse("1" + New String("0", digs))
bs >>= 1
bs += 4
Dim st = bs.ToString
digs -= 1
Return String.Format("{0}.{1}", st(0), st.Substring(1, digs))
End Function
Function DivIt(a As BI, b As BI, digs As Integer) As String
REM performs division
Dim al = a.ToString.Length
Dim bl = b.ToString.Length
digs += 1
a *= BI.Pow(10, digs << 1)
b *= BI.Pow(10, digs)
Dim s = (a / b + 5).ToString
digs -= 1
Return s(0) + "." + s.Substring(1, digs)
End Function
REM custom formatting
Function Joined(x() As BI) As String
Dim wids() = {1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}
Dim res = ""
For i = 0 To x.Length - 1
res += String.Format("{0," + (-wids(i)).ToString + "} ", x(i))
Next
Return res
End Function
Sub Main()
REM calculates and checks each "metal"
Console.WriteLine("Metal B Sq.Rt Iters /---- 32 decimal place value ----\\ Matches Sq.Rt Calc")
Dim t = ""
Dim n As BI
Dim nm1 As BI
Dim k As Integer
Dim j As Integer
For b = 0 To 9
Dim lst(14) As BI
lst(0) = 1
lst(1) = 1
For i = 2 To 14
lst(i) = b * lst(i - 1) + lst(i - 2)
Next
REM since all the iterations (except Pt) are > 15, continue iterating from the end of the list of 15
n = lst(14)
nm1 = lst(13)
k = 0
j = 13
While k = 0
Dim lt = t
t = DivIt(n, nm1, 32)
If lt = t Then
k = If(b = 0, 1, j)
End If
Dim onn = n
n = b * n + nm1
nm1 = onn
j += 1
End While
Console.WriteLine("{0,4} {1} {2,2} {3, 2} {4} {5}" + vbNewLine + "{6,19} {7}", "Pt Au Ag CuSn Cu Ni Al Fe Sn Pb".Split(" ")(b), b, b * b + 4, k, t, t = DoOne(b, 32), "", Joined(lst))
Next
REM now calculate and check big one
n = 1
nm1 = 1
k = 0
j = 1
While k = 0
Dim lt = t
t = DivIt(n, nm1, 256)
If lt = t Then
k = j
End If
Dim onn = n
n += nm1
nm1 = onn
j += 1
End While
Console.WriteLine()
Console.WriteLine("Au to 256 digits:")
Console.WriteLine(t)
Console.WriteLine("Iteration count: {0} Matched Sq.Rt Calc: {1}", k, t = DoOne(1, 256))
End Sub
End Module
- Output:
Metal B Sq.Rt Iters /---- 32 decimal place value ----\\ Matches Sq.Rt Calc Pt 0 4 1 1.00000000000000000000000000000000 True 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Au 1 5 78 1.61803398874989484820458683436564 True 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 Ag 2 8 44 2.41421356237309504880168872420970 True 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 CuSn 3 13 34 3.30277563773199464655961063373525 True 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 Cu 4 20 28 4.23606797749978969640917366873128 True 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 Ni 5 29 25 5.19258240356725201562535524577016 True 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 Al 6 40 23 6.16227766016837933199889354443272 True 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 Fe 7 53 22 7.14005494464025913554865124576352 True 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 Sn 8 68 20 8.12310562561766054982140985597408 True 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 Pb 9 85 20 9.10977222864644365500113714088140 True 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 Au to 256 digits: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144 Iteration count: 616 Matched Sq.Rt Calc: True
Wren
import "/big" for BigInt, BigRat
import "/fmt" for Fmt
var names = ["Platinum", "Golden", "Silver", "Bronze", "Copper","Nickel", "Aluminium", "Iron", "Tin", "Lead"]
var lucas = Fn.new { |b|
Fmt.print("Lucas sequence for $s ratio, where b = $d:", names[b], b)
System.write("First 15 elements: ")
var x0 = 1
var x1 = 1
Fmt.write("$d, $d", x0, x1)
for (i in 1..13) {
var x2 = b*x1 + x0
Fmt.write(", $d", x2)
x0 = x1
x1 = x2
}
System.print()
}
var metallic = Fn.new { |b, dp|
var x0 = BigInt.one
var x1 = BigInt.one
var x2 = BigInt.zero
var bb = BigInt.new(b)
var ratio = BigRat.new(BigInt.one, BigInt.one)
var iters = 0
var prev = ratio.toDecimal(dp)
while (true) {
iters = iters + 1
x2 = bb*x1 + x0
ratio = BigRat.new(x2, x1)
var curr = ratio.toDecimal(dp)
if (prev == curr) {
var plural = (iters == 1) ? " " : "s"
Fmt.print("Value to $d dp after $2d iteration$s: $s\n", dp, iters, plural, curr)
return
}
prev = curr
x0 = x1
x1 = x2
}
}
for (b in 0..9) {
lucas.call(b)
metallic.call(b, 32)
}
System.print("Golden ratio, where b = 1:")
metallic.call(1, 256)
- Output:
Lucas sequence for Platinum ratio, where b = 0: First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Value to 32 dp after 1 iteration : 1 Lucas sequence for Golden ratio, where b = 1: First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 Value to 32 dp after 78 iterations: 1.61803398874989484820458683436564 Lucas sequence for Silver ratio, where b = 2: First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243 Value to 32 dp after 44 iterations: 2.41421356237309504880168872420970 Lucas sequence for Bronze ratio, where b = 3: First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564 Value to 32 dp after 34 iterations: 3.30277563773199464655961063373525 Lucas sequence for Copper ratio, where b = 4: First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141 Value to 32 dp after 28 iterations: 4.23606797749978969640917366873128 Lucas sequence for Nickel ratio, where b = 5: First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686 Value to 32 dp after 25 iterations: 5.19258240356725201562535524577016 Lucas sequence for Aluminium ratio, where b = 6: First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775 Value to 32 dp after 23 iterations: 6.16227766016837933199889354443272 Lucas sequence for Iron ratio, where b = 7: First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408 Value to 32 dp after 22 iterations: 7.14005494464025913554865124576352 Lucas sequence for Tin ratio, where b = 8: First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449 Value to 32 dp after 20 iterations: 8.12310562561766054982140985597408 Lucas sequence for Lead ratio, where b = 9: First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226 Value to 32 dp after 20 iterations: 9.10977222864644365500113714088140 Golden ratio, where b = 1: Value to 256 dp after 615 iterations: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
zkl
GNU Multiple Precision Arithmetic Library
var [const] BI=Import("zklBigNum"); // libGMP
fcn lucasSeq(b){
Walker.zero().tweak('wrap(xs){
xm2,xm1 := xs; // x[n-2], x[n-1]
xn:=xm1*b + xm2;
xs.append(xn).del(0);
xn
}.fp(L(BI(1),BI(1)))).push(1,1) // xn can get big so use BigInts
}
fcn metallicRatio(lucasSeq,digits=32,roundup=True){ #-->(String,num iterations)
bige:=BI("1e"+(digits+1)); # x[n-1]*bige*b / x[n-2] to get our digits from Ints
a,b,mr := lucasSeq.next(), lucasSeq.next(), (bige*b).div(a);
do(20_000){ // limit iterations
c,mr2 := lucasSeq.next(), (bige*c).div(b);
if(mr==mr2){
mr=mr2.add(5*roundup).div(10).toString();
return(String(mr[0],".",mr.del(0)),
lucasSeq.idx); // idx ignores push(), ie first 2 terms
}
b,mr = c,mr2;
}
}
metals:="Platinum Golden Silver Bronze Copper Nickel Aluminum Iron Tin Lead";
foreach metal in (metals.split(" ")){ n:=__metalWalker.idx;
println("\nLucas sequence for %s ratio; where b = %d:".fmt(metal,n));
println("First 15 elements: ",lucasSeq(n).walk(15).concat(" "));
mr,i := metallicRatio(lucasSeq(n));
println("Approximated value: %s - Reached after ~%d iterations.".fmt(mr,i));
}
- Output:
Lucas sequence for Platinum ratio; where b = 0: First 15 elements: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Approximated value: 1.00000000000000000000000000000000 - Reached after ~0 iterations. Lucas sequence for Golden ratio; where b = 1: First 15 elements: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 Approximated value: 1.61803398874989484820458683436564 - Reached after ~81 iterations. Lucas sequence for Silver ratio; where b = 2: First 15 elements: 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 Approximated value: 2.41421356237309504880168872420970 - Reached after ~45 iterations. Lucas sequence for Bronze ratio; where b = 3: First 15 elements: 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 Approximated value: 3.30277563773199464655961063373525 - Reached after ~34 iterations. Lucas sequence for Copper ratio; where b = 4: First 15 elements: 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 Approximated value: 4.23606797749978969640917366873128 - Reached after ~28 iterations. Lucas sequence for Nickel ratio; where b = 5: First 15 elements: 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 Approximated value: 5.19258240356725201562535524577016 - Reached after ~25 iterations. Lucas sequence for Aluminum ratio; where b = 6: First 15 elements: 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 Approximated value: 6.16227766016837933199889354443272 - Reached after ~22 iterations. Lucas sequence for Iron ratio; where b = 7: First 15 elements: 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 Approximated value: 7.14005494464025913554865124576352 - Reached after ~21 iterations. Lucas sequence for Tin ratio; where b = 8: First 15 elements: 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 Approximated value: 8.12310562561766054982140985597408 - Reached after ~20 iterations. Lucas sequence for Lead ratio; where b = 9: First 15 elements: 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 Approximated value: 9.10977222864644365500113714088140 - Reached after ~19 iterations.
println("Golden ratio (B==1) to 256 digits:");
mr,i := metallicRatio(lucasSeq(1),256);
println("Approximated value: %s\nReached after ~%d iterations.".fmt(mr,i));
- Output:
Golden ratio (b==1) to 256 digits: Approximated value: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144 Reached after ~616 iterations.