Pascal's triangle: Difference between revisions
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def prettytri( |
def prettytri(n:Int) = (1 to n) foreach {i=>print(" "*(n-i)); tri(i) map (c=>print(c+" ")); println} |
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prettytri(5) |
prettytri(5) |
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Revision as of 18:01, 3 December 2009
You are encouraged to solve this task according to the task description, using any language you may know.
Pascal's triangle is an interesting math concept. Its first few rows look like this:
1 1 1 1 2 1 1 3 3 1
where each element of each row is either 1 or the sum of the two elements right above it. For example, the next row would be 1 (since the first element of each row doesn't have two elements above it), 4 (1 + 3), 6 (3 + 3), 4 (3 + 1), and 1 (since the last element of each row doesn't have two elements above it). Each row n (starting with row 0 at the top) shows the coefficients of the binomial expansion of (x + y)n.
Write a function that prints out the first n rows of the triangle (with f(1) yielding the row consisting of only the element 1). This can be done either by summing elements from the previous rows or using a binary coefficient or combination function. Behavior for n <= 0 does not need to be uniform, but should be noted.
Ada
<lang ada>with Ada.Integer_Text_Io; use Ada.Integer_Text_Io; with Ada.Text_Io; use Ada.Text_Io;
procedure Pascals_Triangle is
type Row is array(Positive range <>) of Integer;
type Row_Access is access Row;
type Triangle is array(Positive range <>) of Row_Access;
function General_Triangle(Depth : Positive) return Triangle is
Result : Triangle(1..Depth);
begin
for I in Result'range loop
Result(I) := new Row(1..I);
for J in 1..I loop
if J = Result(I)'First or else J = Result(I)'Last then
Result(I)(J) := 1;
else
Result(I)(J) := Result(I - 1)(J - 1) + Result(I - 1)(J);
end if;
end loop;
end loop;
return Result;
end General_Triangle;
procedure Print(Item : Triangle) is
begin
for I in Item'range loop
for J in 1..I loop
Put(Item => Item(I)(J), Width => 3);
end loop;
New_Line;
end loop;
end Print;
begin
Print(General_Triangle(7));
end Pascals_Triangle;</lang>
ALGOL 68
<lang algol68>PRIO MINLWB = 8, MAXUPB = 8; OP MINLWB = ([]INT a,b)INT: (LWB a<LWB b|LWB a|LWB b),
MAXUPB = ([]INT a,b)INT: (UPB a>UPB b|UPB a|UPB b);
OP + = ([]INT a,b)[]INT:(
[a MINLWB b:a MAXUPB b]INT out; FOR i FROM LWB out TO UPB out DO out[i]:= 0 OD; out[LWB a:UPB a] := a; FOR i FROM LWB b TO UPB b DO out[i]+:= b[i] OD; out
);
INT width = 4, stop = 9; FORMAT centre = $n((stop-UPB row+1)*width OVER 2)(q)$;
FLEX[1]INT row := 1; # example of rowing # FOR i WHILE
printf((centre, $g(-width)$, row, $l$));
- WHILE # i < stop DO
row := row[AT 1] + row[AT 2]
OD</lang> Output:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
AutoHotkey
ahk forum: discussion <lang AutoHotkey>n := 8, p0 := "1" ; 1+n rows of Pascal's triangle Loop %n% {
p := "p" A_Index, %p% := v := 1, q := "p" A_Index-1
Loop Parse, %q%, %A_Space%
If (A_Index > 1)
%p% .= " " v+A_LoopField, v := A_LoopField
%p% .= " 1"
}
; Triangular Formatted output
VarSetCapacity(tabs,n,Asc("`t")) t .= tabs "`t1" Loop %n% {
t .= "`n" SubStr(tabs,A_Index)
Loop Parse, p%A_Index%, %A_Space%
t .= A_LoopField "`t`t"
} Gui Add, Text,, %t% ; Show result in a GUI Gui Show Return
GuiClose:
ExitApp</lang>
AWK
<lang awk>$ awk 'BEGIN{for(i=0;i<6;i++){c=1;r=c;for(j=0;j<i;j++){c*=(i-j)/(j+1);r=r" "c};print r}}' 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1</lang>
BASIC
Summing from Previous Rows
This implementation uses an array to store one row of the triangle. DIM initializes the array values to zero. For first row, "1" is then stored in the array. To calculate values for next row, the value in cell (i-1) is added to each cell (i). This summing is done from right to left so that it can be done on-place, without using a tmp buffer. Because of symmetry, the values can be displayed from left to right.
Space for max 5 digit numbers is reserved when formatting the display. The maximum size of triangle is 100 rows, but in practice it is limited by screen space. If the user enters value less than 1, the first row is still always displayed.
<lang freebasic>DIM i AS Integer DIM row AS Integer DIM nrows AS Integer DIM values(100) AS Integer
INPUT "Number of rows: "; nrows values(1) = 1 PRINT TAB((nrows)*3);" 1" FOR row = 2 TO nrows
PRINT TAB((nrows-row)*3+1);
FOR i = row TO 1 STEP -1
values(i) = values(i) + values(i-1)
PRINT USING "##### "; values(i);
NEXT i
PRINT
NEXT row</lang>
Using binary coefficients
This implementation does not need an array to store values for each row, but the calculation is slightly more complex.
If the user enters value less than 1, nothing is displayed. <lang freebasic>DIM c AS Integer DIM i AS Integer DIM row AS Integer DIM nrows AS Integer
INPUT "Number of rows: "; nrows FOR row = 0 TO nrows-1
c = 1 PRINT TAB((nrows-row)*3); FOR i = 0 TO row
PRINT USING "##### "; c;
c = c * (row - i) / (i+1) NEXT i PRINT
NEXT row</lang>
Note: Unlike most Basic implementations, FreeBASIC requires that all variables have been declared before use (but this can be changed with compiler options).
C
<lang c>#include <stdio.h>
void pascaltriangle(unsigned int n) {
unsigned int c, i, j, k;
for(i=0; i < n; i++) {
c = 1;
for(j=1; j <= 2*(n-1-i); j++) printf(" ");
for(k=0; k <= i; k++) {
printf("%3d ", c);
c = c * (i-k)/(k+1);
}
printf("\n");
}
}
int main() {
pascaltriangle(8); return 0;
}</lang>
C++
<lang cpp>#include <iostream>
- include <algorithm>
- include <vector>
- include <iterator>
void genPyrN(int rows) {
if (rows < 0) return;
// save the last row here
std::vector<int> last(1, 1);
std::cout << last[0] << std::endl;
for (int i = 1; i <= rows; i++) {
// work on the next row
std::vector<int> thisRow;
thisRow.reserve(i+1);
thisRow.push_back(last.front()); // beginning of row
std::transform(last.begin(), last.end()-1, last.begin()+1, std::back_inserter(thisRow), std::plus<int>()); // middle of row
thisRow.push_back(last.back()); // end of row
for (int j = 0; j <= i; j++)
std::cout << thisRow[j] << " ";
std::cout << std::endl;
last.swap(thisRow); }
}</lang>
C#
Produces no output when n is less than or equal to zero.
<lang csharp>using System;
namespace RosettaCode {
class PascalsTriangle {
public void CreateTriangle(int n) {
if (n > 0) {
for (int i = 0; i < n; i++) {
int c = 1;
Console.Write(" ".PadLeft(2 * (n - 1 - i)));
for (int k = 0; k <= i; k++) {
Console.Write("{0}", c.ToString().PadLeft(3));
c = c * (i - k) / (k + 1);
}
Console.WriteLine();
}
}
}
}
}</lang>
Clojure
For n < 1, prints nothing, always returns nil. Copied from the Common Lisp implementation below, but with local functions and explicit tail-call-optimized recursion (recur). <lang lisp>(defn pascal [n]
(let [newrow (fn newrow [lst ret]
(if lst
(recur (rest lst)
(conj ret (+ (first lst) (or (second lst) 0))))
ret))
genrow (fn genrow [n lst]
(when (< 0 n)
(do (println lst)
(recur (dec n) (conj (newrow lst []) 1)))))]
(genrow n [1])))
(pascal 4)</lang>
Common Lisp
To evaluate, call (pascal n). For n < 1, it simply returns nil.
<lang lisp>(defun pascal (n)
(genrow n '(1)))
(defun genrow (n l)
(when (< 0 n)
(print l)
(genrow (1- n) (cons 1 (newrow l)))))
(defun newrow (l)
(if (> 2 (length l))
'(1)
(cons (+ (car l) (cadr l)) (newrow (cdr l)))))</lang>
D
There is similarity between Pascal's triangle and Sierpinski triangle. Their difference are the initial line and the operation that act on the line element to produce next line. The following is a generic pascal's triangle implementation for positive number of lines output (n). <lang d>import std.stdio, std.string, std.format : fmx = format ;
string Pascal(alias dg, T, T initValue)(int n) {
string output ;
void append(T[] l) {
output ~= repeat(" ", (n - l.length + 1)*2) ;
foreach(e; l) output ~= fmx("%4s", fmx("%4s",e)) ;
output ~= "\n" ;
}
if(n > 0) {
T[][] lines = initValue ;
append(lines[0]) ;
for(int i = 1 ; i < n ; i++) {
lines ~= lines[i-1] ~ initValue ; // length + 1
for(int j = 1 ; j < lines[i-1].length ; j++)
lines[i][j] = dg(lines[i-1][j], lines[i-1][j-1]) ;
append(lines[i]) ;
}
}
return output ;
}
string delegate(int n) GenericPascal(alias dg, T, T initValue)(){
mixin Pascal!(dg, T, initValue) ; return &Pascal ;
}
char xor(char a, char b) { return a == b ? '_' : '*' ; }
void main() {
auto pascal = GenericPascal!((int a, int b) { return a + b; }, int , 1 ) ;
auto sierpinski = GenericPascal!(xor, char, '*') ;
foreach(i ; [1,5,9]) writef(pascal(i)) ; // an order 4 sierpinski triangle is a 2^4 lines generic pascal triangle with xor operation foreach(i ; [16]) writef(sierpinski(i)) ;
}</lang>
E
So as not to bother with text layout, this implementation generates a HTML fragment. It uses a single mutable array, appending one 1 and adding to each value the preceding value.
<lang e>def pascalsTriangle(n, out) {
def row := [].diverge(int)
out.print("
") for y in 1..n { out.print("") row.push(1) def skip := n - y if (skip > 0) { out.print(``) }
for x => v in row {
out.print(``)
}
for i in (1..!y).descending() {
row[i] += row[i - 1]
}
out.println("")
}
out.print("| $v |
")
}</lang>
<lang e>def out := <file:triangle.html>.textWriter() try {
pascalsTriangle(15, out)
} finally {
out.close()
} makeCommand("yourFavoriteWebBrowser")("triangle.html")</lang>
Factor
This implementation works by summing the previous line content. Result for n < 1 is the same as for n == 1.
<lang factor>USING: grouping kernel math sequences ;
- (pascal) ( seq -- newseq )
dup peek 0 prefix 0 suffix 2 <clumps> [ sum ] map suffix ;
- pascal ( n -- seq )
1 - { { 1 } } swap [ (pascal) ] times ;</lang>
It works as:
<lang factor>5 pascal . { { 1 } { 1 1 } { 1 2 1 } { 1 3 3 1 } { 1 4 6 4 1 } }</lang>
Forth
<lang forth>: init ( n -- )
here swap cells erase 1 here ! ;
- .line ( n -- )
cr here swap 0 do dup @ . cell+ loop drop ;
- next ( n -- )
here swap 1- cells here + do i @ i cell+ +! -1 cells +loop ;
- pascal ( n -- )
dup init 1 .line 1 ?do i next i 1+ .line loop ;</lang>
Fortran
Prints nothing for n<=0. Output formatting breaks down for n>20 <lang fortran>PROGRAM Pascals_Triangle
CALL Print_Triangle(8)
END PROGRAM Pascals_Triangle
SUBROUTINE Print_Triangle(n)
IMPLICIT NONE INTEGER, INTENT(IN) :: n INTEGER :: c, i, j, k, spaces
DO i = 0, n-1
c = 1
spaces = 3 * (n - 1 - i)
DO j = 1, spaces
WRITE(*,"(A)", ADVANCE="NO") " "
END DO
DO k = 0, i
WRITE(*,"(I6)", ADVANCE="NO") c
c = c * (i - k) / (k + 1)
END DO
WRITE(*,*)
END DO
END SUBROUTINE Print_Triangle</lang>
Groovy
Recursive
In the spirit of the Haskell "think in whole lists" solution here is a list-driven, minimalist solution: <lang groovy>def pascal = { n -> (n <= 1) ? [1] : GroovyCollections.transpose([[0] + pascal(n - 1), pascal(n - 1) + [0]]).collect { it.sum() } }</lang> However, this solution is horribly inefficient (O(n**2)). It slowly grinds to a halt on a reasonably powerful PC after about line 25 of the triangle.
Test program: <lang groovy>def count = 15 (1..count).each { n ->
printf ("%2d:", n); (0..(count-n)).each { print " " }; pascal(n).each{ printf("%6d ", it) }; println ""
}</lang>
Output:
1: 1 2: 1 1 3: 1 2 1 4: 1 3 3 1 5: 1 4 6 4 1 6: 1 5 10 10 5 1 7: 1 6 15 20 15 6 1 8: 1 7 21 35 35 21 7 1 9: 1 8 28 56 70 56 28 8 1 10: 1 9 36 84 126 126 84 36 9 1 11: 1 10 45 120 210 252 210 120 45 10 1 12: 1 11 55 165 330 462 462 330 165 55 11 1 13: 1 12 66 220 495 792 924 792 495 220 66 12 1 14: 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 15: 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1
Haskell
An approach using the "think in whole lists" principle: Each row in the triangle can be calculated from the previous row by adding a shifted version of itself to it, keeping the ones at the ends. The Prelude function zipWith can be used to add two lists, but it won't keep the old values when one list is shorter. So we need a similar function
<lang haskell>zapWith :: (a -> a -> a) -> [a] -> [a] -> [a] zapWith f xs [] = xs zapWith f [] ys = ys zapWith f (x:xs) (y:ys) = f x y : zapWith f xs ys</lang>
Now we can shift a list and add it to itself, extending it by keeping the ends:
<lang haskell>extendWith f [] = [] extendWith f xs@(x:ys) = x : zapWith f xs ys</lang>
And for the whole (infinite) triangle, we just iterate this operation, starting with the first row:
<lang haskell>pascal = iterate (extendWith (+)) [1]</lang>
For the first n rows, we just take the first n elements from this list, as in
<lang haskell>*Main> take 6 pascal [[1],[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1],[1,5,10,10,5,1]]</lang>
A shorter approach, plagiarized from [1] <lang haskell>-- generate next row from current row nextRow row = zipWith (+) ([0] ++ row) (row ++ [0])
-- returns the first n rows pascal = iterate nextRow [1]</lang>
J
solution
<lang j>!/~@i. N</lang>
explanation
Of course, the triangle itself is simply the table of number-of-combinations, for the first N non-negative integers.
That is, C(n,k) for all n,k ∈ [0 .. n). J's notation for C(n,k) is k ! n (mnemonic: combinations are closely related to factorials, which are denoted by ! in math).
So, for example, the number of ways to choose a poker hand (5 cards from the deck of 52): <lang j>5!52
2598960</lang>
So ! is the mathematical choose function. What of /~@i.? Well, you can think of /~ as "table of" and @i. "the first N non-negative integers (i.e. 0 .. N-1)".
So, for example, here's the multiplication table you had to memorize in first grade: <lang j>*/~@i. 10
0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 16 18 0 3 6 9 12 15 18 21 24 27 0 4 8 12 16 20 24 28 32 36 0 5 10 15 20 25 30 35 40 45 0 6 12 18 24 30 36 42 48 54 0 7 14 21 28 35 42 49 56 63 0 8 16 24 32 40 48 56 64 72 0 9 18 27 36 45 54 63 72 81</lang>
and here's the addition table for 0 to 4 <lang j>+/~@i. 4 0 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6</lang>
Similarly, !/~@i. is the number-of-combinations table, or the "choose" table: <lang j>!/~@i. 5 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1</lang>
Of course, to format it nicely, you need to do a little more work: <lang j>([:'0'&=`(,:&' ')} -@|. |."_1 [: ":@|: !/~)@i. 5
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1</lang>
(I won't bother explaining the formatting.)
IDL
<lang IDL>Pro Pascal, n
- n is the number of lines of the triangle to be displayed
r=[1] print, r for i=0, (n-2) do begin pascalrow,r endfor
End
Pro PascalRow, r
for i=0,(n_elements(r)-2) do begin r[i]=r[i]+r[i+1] endfor
r= [1, r] print, r
End</lang>
Java
Summing from Previous Rows
<lang java>import java.util.ArrayList; ...//class definition, etc. public static void genPyrN(int rows){ if(rows < 0) return; //save the last row here ArrayList<Integer> last = new ArrayList<Integer>(); last.add(1); System.out.println(last); for(int i= 1;i <= rows;++i){ //work on the next row ArrayList<Integer> thisRow= new ArrayList<Integer>(); thisRow.add(last.get(0)); //beginning for(int j= 1;j < i;++j){//loop the number of elements in this row //sum from the last row thisRow.add(last.get(j - 1) + last.get(j)); } thisRow.add(last.get(i)); //end last= thisRow;//save this row System.out.println(thisRow); } }</lang>
Combinations
This method is limited to 21 rows because of the limits of long. Calling pas with an argument of 22 or above will cause intermediate math to wrap around and give false answers. <lang java>public class Pas{ public static void main(String[] args){ //usage pas(20); }
public static void pas(int rows){ for(int i = 0; i < rows; i++){ for(int j = 0; j <= i; j++){ System.out.print(ncr(i, j) + " "); } System.out.println(); } }
public static long ncr(int n, int r){ return fact(n) / (fact(r) * fact(n - r)); }
public static long fact(int n){ long ans = 1; for(int i = 2; i <= n; i++){ ans *= i; } return ans; } }</lang>
Logo
<lang logo>to pascal :n
if :n = 1 [output [1]] localmake "a pascal :n-1 output (sentence first :a (map "sum butfirst :a butlast :a) last :a)
end
for [i 1 10] [print pascal :i]</lang>
Metafont
(The formatting starts to be less clear when numbers start to have more than two digits)
<lang metafont>vardef bincoeff(expr n, k) = save ?; ? := (1 for i=(max(k,n-k)+1) upto n: * i endfor )
/ (1 for i=2 upto min(k, n-k): * i endfor); ?
enddef;
def pascaltr expr c =
string s_;
for i := 0 upto (c-1):
s_ := "" for k=0 upto (c-i): & " " endfor;
s_ := s_ for k=0 upto i: & decimal(bincoeff(i,k))
& " " if bincoeff(i,k)<9: & " " fi endfor;
message s_;
endfor
enddef;
pascaltr(4); end</lang>
Nial
Like J
(pascal.nial) <lang nial>factorial is recur [ 0 =, 1 first, pass, product, -1 +] combination is fork [ > [first, second], 0 first,
/ [factorial second, * [factorial - [second, first], factorial first] ]
] pascal is transpose each combination cart [pass, pass] tell</lang> Using it <lang nial>|loaddefs 'pascal.nial' |pascal 5</lang>
OCaml
<lang ocaml>(* generate next row from current row *) let next_row row =
List.map2 (+) ([0] @ row) (row @ [0])
(* returns the first n rows *) let pascal n =
let rec loop i row = if i = n then [] else row :: loop (i+1) (next_row row) in loop 0 [1]</lang>
Octave
<lang octave>function pascaltriangle(h)
for i = 0:h-1
for k = 0:h-i
printf(" ");
endfor
for j = 0:i
printf("%3d ", bincoeff(i, j));
endfor
printf("\n");
endfor
endfunction
pascaltriangle(4);</lang>
Perl
<lang perl>sub pascal
- Prints out $n rows of Pascal's triangle. It returns undef for
- failure and 1 for success.
{my $n = shift;
$n < 1 and return undef;
print "1\n";
$n == 1 and return 1;
my @last = (1);
foreach my $row (1 .. $n - 1)
{my @this = map {$last[$_] + $last[$_ + 1]} 0 .. $row - 2;
@last = (1, @this, 1);
print join(' ', @last), "\n";}
return 1;}</lang>
Python
<lang python>def pascal(n):
"""Prints out n rows of Pascal's triangle.
It returns False for failure and True for success."""
row = [1,]
k = [0,]
for x in range(max(n,0)):
print row
row=[l+r for l,r in zip(row+k,k+row)]
return n>=1</lang>
R
<lang R>pascalTriangle <- function(h) {
for(i in 0:(h-1)) {
s <- ""
for(k in 0:(h-i)) s <- paste(s, " ", sep="")
for(j in 0:i) {
s <- paste(s, sprintf("%3d ", choose(i, j)), sep="")
}
print(s)
}
}</lang>
RapidQ
Summing from Previous Rows
The main difference to BASIC implementation is the output formatting. RapidQ does not support PRINT USING. Instead, function FORMAT$() is used. TAB() is not supported, so SPACE$() was used instead.
Another difference is that in RapidQ, DIM does not clear array values to zero. But if dimensioning is done with DEF..., you can give the initial values in curly braces. If less values are given than there are elements in the array, the remaining positions are initialized to zero. RapidQ does not require simple variables to be declared before use.
<lang rapidq>DEFINT values(100) = {0,1}
INPUT "Number of rows: "; nrows PRINT SPACE$((nrows)*3);" 1" FOR row = 2 TO nrows
PRINT SPACE$((nrows-row)*3+1);
FOR i = row TO 1 STEP -1
values(i) = values(i) + values(i-1)
PRINT FORMAT$("%5d ", values(i));
NEXT i
PRINT
NEXT row</lang>
Using binary coefficients
<lang rapidq>INPUT "Number of rows: "; nrows FOR row = 0 TO nrows-1
c = 1
PRINT SPACE$((nrows-row)*3);
FOR i = 0 TO row
PRINT FORMAT$("%5d ", c);
c = c * (row - i) / (i+1)
NEXT i
PRINT
NEXT row</lang>
Ruby
<lang ruby>def pascal(n = 1)
return if n < 1
# set up a new array of arrays with the first value p = 1 # for n - 1 number of times, (n - 1).times do |i|
# inject a new array starting with [1]
p << p[i].inject([1]) do |result, elem|
# if we've reached the end, tack on a 1.
# else, tack on current elem + previous elem
if p[i].length == result.length
result << 1
else
result << elem + p[i][result.length]
end
end
end
# and return the triangle.
p
end</lang>
Scala
Simple recursive row definition: <lang scala> def tri(row:Int):List[Int] = { row match {
case 1 => List(1)
case n:Int => List(1) ::: ((tri(n-1) zip tri(n-1).tail) map {case (a,b) => a+b}) ::: List(1)
}
} </lang> Funtion to pretty print n rows: <lang scala> def prettytri(n:Int) = (1 to n) foreach {i=>print(" "*(n-i)); tri(i) map (c=>print(c+" ")); println} prettytri(5) </lang> Outputs:
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1
Seed7
<lang seed7>$ include "seed7_05.s7i";
const proc: main is func
local
var integer: numRows is 0;
var array integer: values is [] (0, 1);
var integer: row is 0;
var integer: index is 0;
begin
write("Number of rows: ");
readln(numRows);
writeln("1" lpad succ(numRows) * 3);
for row range 2 to numRows do
write("" lpad (numRows - row) * 3);
values &:= [] 0;
for index range succ(row) downto 2 do
values[index] +:= values[pred(index)];
write(" " <& values[index] lpad 5);
end for;
writeln;
end for;
end func;</lang>
Tcl
Summing from Previous Rows
<lang tcl>proc pascal_iterative n {
if {$n < 1} {error "undefined behaviour for n < 1"}
set row [list 1]
lappend rows $row
set i 1
while {[incr i] <= $n} {
set prev $row
set row [list 1]
for {set j 1} {$j < [llength $prev]} {incr j} {
lappend row [expr {[lindex $prev [expr {$j - 1}]] + [lindex $prev $j]}]
}
lappend row 1
lappend rows $row
}
return $rows
}
puts [join [pascal_iterative 6] \n]</lang>
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1
Using binary coefficients
<lang tcl>proc pascal_coefficients n {
if {$n < 1} {error "undefined behaviour for n < 1"}
for {set i 0} {$i < $n} {incr i} {
set c 1
set row [list $c]
for {set j 0} {$j < $i} {incr j} {
set c [expr {$c * ($i - $j) / ($j + 1)}]
lappend row $c
}
lappend rows $row
}
return $rows
}
puts [join [pascal_coefficients 6] \n]</lang>
Combinations
, however thanks to Tcl 8.5's arbitrary precision integer arithmetic, this solution is not limited to a couple of dozen rows. Uses a caching factorial calculator to improve performance.
<lang tcl>package require Tcl 8.5
proc pascal_combinations n {
if {$n < 1} {error "undefined behaviour for n < 1"}
for {set i 0} {$i < $n} {incr i} {
set row [list]
for {set j 0} {$j <= $i} {incr j} {
lappend row [C $i $j]
}
lappend rows $row
}
return $rows
}
proc C {n k} {
expr {[ifact $n] / ([ifact $k] * [ifact [expr {$n - $k}]])}
}
set fact_cache {1 1} proc ifact n {
global fact_cache
if {$n < [llength $fact_cache]} {
return [lindex $fact_cache $n]
}
set i [expr {[llength $fact_cache] - 1}]
set sum [lindex $fact_cache $i]
while {$i < $n} {
incr i
set sum [expr {$sum * $i}]
lappend fact_cache $sum
}
return $sum
}
puts [join [pascal_combinations 6] \n]</lang>
Comparing Performance
<lang tcl>set n 100 puts "calculate $n rows:" foreach proc {pascal_iterative pascal_coefficients pascal_combinations} {
puts "$proc: [time [list $proc $n] 100]"
}</lang> outputs
calculate 100 rows: pascal_iterative: 2800.14 microseconds per iteration pascal_coefficients: 8760.98 microseconds per iteration pascal_combinations: 38176.66 microseconds per iteration
Ursala
Zero maps to the empty list. Negatives are inexpressible. This solution uses a library function for binomial coefficients. <lang Ursala>#import std
- import nat
pascal = choose**ziDS+ iota*t+ iota+ successor</lang> This solution uses direct summation. The algorithm is to insert zero at the head of a list (initially the unit list <1>), zip it with its reversal, map the sum over the list of pairs, iterate n times, and return the trace. <lang Ursala>#import std
- import nat
pascal "n" = (next"n" sum*NiCixp) <1></lang> test program: <lang Ursala>#cast %nLL
example = pascal 10</lang> output:
< <1>, <1,1>, <1,2,1>, <1,3,3,1>, <1,4,6,4,1>, <1,5,10,10,5,1>, <1,6,15,20,15,6,1>, <1,7,21,35,35,21,7,1>, <1,8,28,56,70,56,28,8,1>, <1,9,36,84,126,126,84,36,9,1>>
Vedit macro language
Summing from Previous Rows
Vedit macro language does not have actual arrays (edit buffers are normally used for storing larger amounts of data). However, a numeric register can be used as index to access another numeric register. For example, if #99 contains value 2, then #@99 accesses contents of numeric register #2.
<lang vedit>#100 = Get_Num("Number of rows: ", STATLINE)
- 0=0; #1=1
Ins_Char(' ', COUNT, #100*3-2) Num_Ins(1) for (#99 = 2; #99 <= #100; #99++) {
Ins_Char(' ', COUNT, (#100-#99)*3)
#@99 = 0
for (#98 = #99; #98 > 0; #98--) {
#97 = #98-1 #@98 += #@97 Num_Ins(#@98, COUNT, 6)
} Ins_Newline
}</lang>
Using binary coefficients
<lang vedit>#1 = Get_Num("Number of rows: ", STATLINE) for (#2 = 0; #2 < #1; #2++) {
#3 = 1
Ins_Char(' ', COUNT, (#1-#2-1)*3)
for (#4 = 0; #4 <= #2; #4++) {
Num_Ins(#3, COUNT, 6) #3 = #3 * (#2-#4) / (#4+1)
} Ins_Newline
}</lang>