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# Pell numbers

Pell numbers
You are encouraged to solve this task according to the task description, using any language you may know.

Pell numbers are an infinite sequence of integers that comprise the denominators of the closest rational approximations to the square root of 2 but have many other interesting uses and relationships.

The numerators of each term of rational approximations to the square root of 2 may also be derived from Pell numbers, or may be found by taking half of each term of the related sequence: Pell-Lucas or Pell-companion numbers.

The Pell numbers: 0, 1, 2, 5, 12, 29, 70, etc., are defined by the recurrence relation:

```
P0 = 0;
P1 = 1;
Pn = 2 × Pn-1 + Pn-2;

```

Or, may also be expressed by the closed form formula:

```
Pn = ((1 + √2)n - (1 - √2)n) / (2 × √2);

```

Pell-Lucas or Pell-companion numbers: 2, 2, 6, 14, 34, 82, etc., are defined by a very similar recurrence relation, differing only in the first two terms:

```
Q0 = 2;
Q1 = 2;
Qn = 2 × Qn-1 + Qn-2;

```

Or, may also be expressed by the closed form formula:

```
Qn = (1 + √2)n + (1 - √2)n;

```

or

```
Qn = P2n / Pn;

```

The sequence of rational approximations to the square root of 2 begins:

```
1/1, 3/2, 7/5, 17/12, 41/29, ...

```

Starting from n = 1, for each term, the denominator is Pn and the numerator is Qn / 2 or Pn-1 + Pn.

Pell primes are Pell numbers that are prime. Pell prime indices are the indices of the primes in the Pell numbers sequence. Every Pell prime index is prime, though not every prime index corresponds to a prime Pell number.

If you take the sum S of the first 4n + 1 Pell numbers, the sum of the terms P2n and P2n + 1 will form the square root of S.

For instance, the sum of the Pell numbers up to P5; 0 + 1 + 2 + 5 + 12 + 29 == 49, is the square of P2 + P3 == 2 + 5 == 7. The sequence of numbers formed by the sums P2n + P2n + 1 are known as Newman-Shank-Williams numbers or NSW numbers.

Pell numbers may also be used to find Pythagorean triple near isosceles right triangles; right triangles whose legs differ by exactly 1. E.G.: (3,4,5), (20,21,29), (119,120,169), etc.

For n > 0, each right triangle hypotenuse is P2n + 1. The shorter leg length is the sum of the terms up to P2n + 1. The longer leg length is 1 more than that.

• Find and show at least the first 10 Pell numbers.
• Find and show at least the first 10 Pell-Lucas numbers.
• Use the Pell (and optionally, Pell-Lucas) numbers sequence to find and show at least the first 10 rational approximations to √2 in both rational and decimal representation.
• Find and show at least the first 10 Pell primes.
• Find and show at least the first 10 indices of Pell primes.
• Find and show at least the first 10 Newman-Shank-Williams numbers
• Find and show at least the first 10 Pythagorean triples corresponding to near isosceles right triangles.

## Contents

### FreeBASIC

Translation of: Phix
`#define isOdd(a) (((a) and 1) <> 0) Function isPrime(Byval ValorEval As Integer) As Boolean    If ValorEval < 2 Then Return False    If ValorEval Mod 2 = 0 Then Return ValorEval = 2    If ValorEval Mod 3 = 0 Then Return ValorEval = 3    Dim d As Integer = 5    While d * d <= ValorEval        If ValorEval Mod d = 0 Then Return False Else d += 2        If ValorEval Mod d = 0 Then Return False Else d += 4    Wend     Return TrueEnd Function Dim As Integer nDim As Integer p(0 To 40), pl(0 To 40)p(0)= 0: p(1) = 1pl(0) = 2: pl(1) = 2For n = 2 To 40    p(n) = 2 * p(n-1) + p(n-2)    pl(n) = 2 * pl(n-1) + pl(n-2)Next n Print "First 20 Pell numbers: "For n = 0 To 19 : Print p(n); : Next nPrint !"\n\nFirst 20 Pell-Lucas: "For n = 0 To 19 : Print pl(n); : Next n Print !"\n\nFirst 20 rational approximations of sqrt(2) (" & Str(Sqr(2)) & "): "For n = 1 To 20    Dim As Integer j = pl(n)/2, d = p(n)    Print Using " &/& ~= &"; j; d; j/dNext n Print !"\nFirst 6 Pell primes: [for the limitations of the FB standard library]"Dim as Integer pdx = 2Dim As Byte c = 0Dim As Ulongint ppdx(1 to 20)do    If isPrime(p(pdx)) Then         If isPrime(pdx) Then ppdx(c) = pdx : End If        Print p(pdx)        c += 1     End If    pdx += 1loop until c = 6 Print !"\nIndices of first 6 Pell primes: [for the limitations of the FB standard library]"For n = 0 To 5 : Print " "; ppdx(n); : Next n Dim As Ulongint nsw(0 To 20)For n = 0 To 19    nsw(n) = p(2*n) + p(2*n+1)Next nPrint !"\n\nFirst 20 Newman-Shank-Williams numbers: "For n = 0 To 19 : Print " "; nsw(n); : Next n Print !"\n\nFirst 20 near isosceles right triangles:"Dim As Integer i0 = 0, i1 = 1, i2, t = 1, i = 2, found = 0Do While found < 20    i2 = i1*2 + i0    If isOdd(i) Then        Print Using " [&, &, &]";  t; t+1 ; i2        found += 1    End If    t += i2    i0 = i1 : i1 = i2    i += 1LoopSleep`
Output:
```First 20 Pell numbers:
0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

First 20 Pell-Lucas:
2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

First 20 rational approximations of sqrt(2) (1.414213562373095):
1/1 ~= 1
3/2 ~= 1.5
7/5 ~= 1.4
17/12 ~= 1.416666666666667
41/29 ~= 1.413793103448276
99/70 ~= 1.414285714285714
239/169 ~= 1.414201183431953
577/408 ~= 1.41421568627451
1393/985 ~= 1.414213197969543
3363/2378 ~= 1.41421362489487
8119/5741 ~= 1.414213551646055
19601/13860 ~= 1.414213564213564
47321/33461 ~= 1.41421356205732
114243/80782 ~= 1.414213562427273
275807/195025 ~= 1.414213562363799
665857/470832 ~= 1.41421356237469
1607521/1136689 ~= 1.414213562372821
3880899/2744210 ~= 1.414213562373142
9369319/6625109 ~= 1.414213562373087
22619537/15994428 ~= 1.414213562373096

First 6 Pell primes: [for the limitations of the FB standard library]
2
5
29
5741
33461
44560482149

Indices of first 6 Pell primes: [for the limitations of the FB standard library]
2 3 5 11 13 29

First 20 Newman-Shank-Williams numbers:
1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607

First 20 near isosceles right triangles:
[3, 4, 5]
[20, 21, 29]
[119, 120, 169]
[696, 697, 985]
[4059, 4060, 5741]
[23660, 23661, 33461]
[137903, 137904, 195025]
[803760, 803761, 1136689]
[4684659, 4684660, 6625109]
[27304196, 27304197, 38613965]
[159140519, 159140520, 225058681]
[927538920, 927538921, 1311738121]
[5406093003, 5406093004, 7645370045]
[31509019100, 31509019101, 44560482149]
[183648021599, 183648021600, 259717522849]
[1070379110496, 1070379110497, 1513744654945]
[6238626641379, 6238626641380, 8822750406821]
[36361380737780, 36361380737781, 51422757785981]
[211929657785303, 211929657785304, 299713796309065]
[1235216565974040, 1235216565974041, 1746860020068409]```

## Go

Translation of: Wren
Library: Go-rcu
`package main import (    "fmt"    "math/big"    "rcu") func main() {    p := make([]int64, 40)    p[1] = 1    for i := 2; i < 40; i++ {        p[i] = 2*p[i-1] + p[i-2]    }    fmt.Println("The first 20 Pell numbers are:")    fmt.Println(p[0:20])     q := make([]int64, 40)    q[0] = 2    q[1] = 2    for i := 2; i < 40; i++ {        q[i] = 2*q[i-1] + q[i-2]    }    fmt.Println("\nThe first 20 Pell-Lucas numbers are:")    fmt.Println(q[0:20])     fmt.Println("\nThe first 20 rational approximations of √2 (1.4142135623730951) are:")    for i := 1; i <= 20; i++ {        r := big.NewRat(q[i]/2, p[i])        fmt.Printf("%-17s ≈ %-18s\n", r, r.FloatString(16))    }     fmt.Println("\nThe first 15 Pell primes are:")    p0 := big.NewInt(0)    p1 := big.NewInt(1)    p2 := big.NewInt(0)    two := big.NewInt(2)    indices := make([]int, 15)    for index, count := 2, 0; count < 15; index++ {        p2.Mul(p1, two)        p2.Add(p2, p0)        if rcu.IsPrime(index) && p2.ProbablyPrime(15) {            fmt.Println(p2)            indices[count] = index            count++        }        p0.Set(p1)        p1.Set(p2)    }     fmt.Println("\nIndices of the first 15 Pell primes are:")    fmt.Println(indices)     fmt.Println("\nFirst 20 Newman-Shank_Williams numbers:")    nsw := make([]int64, 20)    for n := 0; n < 20; n++ {        nsw[n] = p[2*n] + p[2*n+1]    }    fmt.Println(nsw)     fmt.Println("\nFirst 20 near isosceles right triangles:")    u0 := 0    u1 := 1    sum := 1    for i := 2; i < 43; i++ {        u2 := u1*2 + u0        if i%2 == 1 {            fmt.Printf("(%d, %d, %d)\n", sum, sum+1, u2)        }        sum += u2        u0 = u1        u1 = u2    }}`
Output:
```The first 20 Pell numbers are:
[0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109]

The first 20 Pell-Lucas numbers are:
[2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638]

The first 20 rational approximations of √2 (1.4142135623730951) are:
1/1               ≈ 1.0000000000000000
3/2               ≈ 1.5000000000000000
7/5               ≈ 1.4000000000000000
17/12             ≈ 1.4166666666666667
41/29             ≈ 1.4137931034482759
99/70             ≈ 1.4142857142857143
239/169           ≈ 1.4142011834319527
577/408           ≈ 1.4142156862745098
1393/985          ≈ 1.4142131979695431
3363/2378         ≈ 1.4142136248948696
8119/5741         ≈ 1.4142135516460547
19601/13860       ≈ 1.4142135642135642
47321/33461       ≈ 1.4142135620573205
114243/80782      ≈ 1.4142135624272734
275807/195025     ≈ 1.4142135623637995
665857/470832     ≈ 1.4142135623746899
1607521/1136689   ≈ 1.4142135623728214
3880899/2744210   ≈ 1.4142135623731420
9369319/6625109   ≈ 1.4142135623730870
22619537/15994428 ≈ 1.4142135623730964

The first 15 Pell primes are:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841

Indices of the first 15 Pell primes are:
[2 3 5 11 13 29 41 53 59 89 97 101 167 181 191]

First 20 Newman-Shank_Williams numbers:
[1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607]

First 20 near isosceles right triangles:
(3, 4, 5)
(20, 21, 29)
(119, 120, 169)
(696, 697, 985)
(4059, 4060, 5741)
(23660, 23661, 33461)
(137903, 137904, 195025)
(803760, 803761, 1136689)
(4684659, 4684660, 6625109)
(27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)
(927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)
(31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)
(1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)
(36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065)
(1235216565974040, 1235216565974041, 1746860020068409)
```

## J

As detailed in the task description, there's a variety of ways to compute these values.

For example:

`nextPell=: , 1 2+/ .*_2&{. NB. pell, list extenderPn=: (%:8) %~(1+%:2)&^ - (1-%:2)&^ NB. pell, closed formQn=: (1+%:2)&^ + (1-%:2)&^         NB. pell lucas, closed formQN=: +: %&Pn ]                     NB. pell lucas, closed formqn=: 2 * (+&Pn <:)                 NB. pell lucas, closed form`

Thus:

`   nextPell^:9(0 1)0 1 2 5 12 29 70 169 408 985 2378   Pn i.110 1 2 5 12 29 70 169 408 985 2378   nextPell^:9(2 2)2 2 6 14 34 82 198 478 1154 2786 6726   Qn i.112 2 6 14 34 82 198 478 1154 2786 6726   QN i.110 2 6 14 34 82 198 478 1154 2786 6726   qn i.112 2 6 14 34 82 198 478 1154 2786 6726`

QN (which is defined as P2n/Pn) doesn't get the first element of the pell lucas sequence right. We could fix this by changing the definition:

`QN=: 2 >. +: %&Pn ]   QN i.112 2 6 14 34 82 198 478 1154 2786 6726`

Continuing... the first ten rational approximations to √2 here would be:

`   }.(%~ _1}. +//[email protected],:~) nextPell^:9(0 1)1 1.5 1.4 1.41667 1.41379 1.41429 1.4142 1.41422 1.41421 1.41421   }.(%~ _1}. +//[email protected],:~) nextPell^:9(0 1x)1 3r2 7r5 17r12 41r29 99r70 239r169 577r408 1393r985 3363r2378`

The first ten pell primes are:

`   10{.(#~ 1&p:)nextPell^:99(0 1x)2 5 29 5741 33461 44560482149 1746860020068409 68480406462161287469 13558774610046711780701 4125636888562548868221559797461449`

Their indices are:

`   10{.I. 1&p:nextPell^:99(0 1x)2 3 5 11 13 29 41 53 59 89`

The NSW numbers are the sums of (non-overlapping) pairs of pell numbers, or:

`   _2 +/\ nextPell^:20(0 1x)1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 `

The first ten pell based pythogorean triples would be:

`   }.(21\$1 0)#|:(}.,~0 1+/+/\@}:)nextPell^:(20)0 1       3        4        5      20       21       29     119      120      169     696      697      985    4059     4060     5741   23660    23661    33461  137903   137904   195025  803760   803761  1136689 4684659  4684660  662510927304196 27304197 38613965`

## Julia

`using Primes function pellnumbers(wanted)    pells = [0, 1]    wanted < 3 && return pells[1:wanted]    while length(pells) < wanted        push!(pells, 2 * pells[end] + pells[end - 1])    end    return pellsend function pelllucasnumbers(wanted)    pelllucas = [2, 2]    wanted < 3 && return pelllucas[1:wanted]    while length(pelllucas) < wanted        push!(pelllucas, 2 * pelllucas[end] + pelllucas[end - 1])    end    return pelllucasend function pellprimes(wanted)    i, lastpell, lastlastpell, primeindices, pellprimes = 1, big"1", big"0", Int[], BigInt[]    while length(primeindices) < wanted        pell = 2 * lastpell + lastlastpell        i += 1        if isprime(pell)            push!(primeindices, i)            push!(pellprimes, pell)        end        lastpell, lastlastpell = pell, lastpell    end    return primeindices, pellprimesend function approximationsqrt2(wanted)    p, q = pellnumbers(wanted + 1), pelllucasnumbers(wanted + 1)    return map(r -> "\$r ≈ \$(Float64(r))", [(q[n] // 2) // p[n] for n in 2:wanted+1])end function newmanshankwilliams(wanted)    pells = pellnumbers(wanted * 2 + 1)    return [pells[2i - 1] + pells[2i] for i in 1:wanted]end function nearisosceles(wanted)    pells = pellnumbers((wanted + 1) * 2 + 1)    return map(x -> (last(x), last(x) + 1, first(x)),       [(pells[2i], sum(pells[1:2i-1])) for i in 2:wanted+1])end function printrows(title, vec, columnsize = 8, columns = 10, rjust=false)    println(title)    for (i, n) in enumerate(vec)        print((rjust ? lpad : rpad)(n, columnsize), i % columns == 0 ? "\n" : "")    end    println()end printrows("Twenty Pell numbers:", pellnumbers(20))printrows("Twenty Pell-Lucas numbers:", pelllucasnumbers(20))printrows("Twenty approximations of √2:", approximationsqrt2(20), 44, 2)pindices, pprimes = pellprimes(15)printrows("Fifteen Pell primes:", pprimes, 90, 1)printrows("Fifteen Pell prime zero-based indices:", pindices, 4, 15)printrows("Twenty Newman-Shank-Williams numbers:", newmanshankwilliams(20), 17, 5)printrows("Twenty near isosceles triangle triplets:", nearisosceles(20), 52, 2) `
Output:
```Twenty Pell numbers:
0       1       2       5       12      29      70      169     408     985
2378    5741    13860   33461   80782   195025  470832  1136689 2744210 6625109

Twenty Pell-Lucas numbers:
2       2       6       14      34      82      198     478     1154    2786
6726    16238   39202   94642   228486  551614  1331714 3215042 7761798 18738638

Twenty approximations of √2:
1//1 ≈ 1.0                                  3//2 ≈ 1.5
7//5 ≈ 1.4                                  17//12 ≈ 1.4166666666666667
41//29 ≈ 1.4137931034482758                 99//70 ≈ 1.4142857142857144
239//169 ≈ 1.4142011834319526               577//408 ≈ 1.4142156862745099
1393//985 ≈ 1.4142131979695431              3363//2378 ≈ 1.4142136248948696
8119//5741 ≈ 1.4142135516460548             19601//13860 ≈ 1.4142135642135643
47321//33461 ≈ 1.4142135620573204           114243//80782 ≈ 1.4142135624272734
275807//195025 ≈ 1.4142135623637995         665857//470832 ≈ 1.4142135623746899
1607521//1136689 ≈ 1.4142135623728214       3880899//2744210 ≈ 1.414213562373142
9369319//6625109 ≈ 1.414213562373087        22619537//15994428 ≈ 1.4142135623730965

Fifteen Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841

Fifteen Pell prime zero-based indices:
2   3   5   11  13  29  41  53  59  89  97  101 167 181 191

Twenty Newman-Shank-Williams numbers:
1                7                41               239              1393
8119             47321            275807           1607521          9369319
54608393         318281039        1855077841       10812186007      63018038201
367296043199     2140758220993    12477253282759   72722761475561   423859315570607

Twenty near isosceles triangle triplets:
(3, 4, 5)                                           (20, 21, 29)
(119, 120, 169)                                     (696, 697, 985)
(4059, 4060, 5741)                                  (23660, 23661, 33461)
(137903, 137904, 195025)                            (803760, 803761, 1136689)
(4684659, 4684660, 6625109)                         (27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)                   (927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)                (31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)          (1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)       (36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065) (1235216565974040, 1235216565974041, 1746860020068409)
```

## Perl

Library: ntheory
`use strict;use warnings;use feature <state say>;use bignum;use ntheory 'is_prime';use List::Util <sum head>;use List::Lazy 'lazy_list'; my \$upto = 20; my \$pell_recur       = lazy_list { state @p = (0, 1); push @p, 2*\$p[1] + \$p[0]; shift @p };my \$pell_lucas_recur = lazy_list { state @p = (2, 2); push @p, 2*\$p[1] + \$p[0]; shift @p }; my @pell;push @pell, \$pell_recur->next() for 1..1500; #wart; my @pell_lucas;push @pell_lucas, \$pell_lucas_recur->next() for 1..\$upto+1; say "First \$upto Pell numbers:";say join ' ', @pell[0..\$upto-1]; say "\nFirst \$upto Pell-Lucas numbers:";say join ' ', @pell_lucas[0..\$upto-1]; say "\nFirst \$upto rational approximations of √2:";say sprintf "%d/%d - %1.16f", \$pell[\$_-1] + \$pell[\$_], \$pell[\$_], (\$pell[\$_-1]+\$pell[\$_])/\$pell[\$_] for 1..\$upto; say "\nFirst \$upto Pell primes:";say join "\n", head \$upto, grep { is_prime \$_ } @pell; say "\nIndices of first \$upto Pell primes:";say join ' ', head \$upto, grep { is_prime(\$pell[\$_]) and \$_ } 0..\$#pell; say "\nFirst \$upto Newman-Shank-Williams numbers:";say join ' ', map { \$pell[2 * \$_] + \$pell[2 * \$_+1] } 0..\$upto-1; say "\nFirst \$upto near isoceles right tringles:";map {    my \$y = 2*\$_ + 1;    my \$x = sum @pell[0..\$y-1];    printf "(%d, %d, %d)\n", \$x, \$x+1, \$pell[\$y]} 1..\$upto;`
Output:
```First 20 Pell numbers:
0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

First 20 Pell-Lucas numbers:
2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

First 20 rational approximations of √2:
1/1 - 1.0000000000000000
3/2 - 1.5000000000000000
7/5 - 1.4000000000000000
17/12 - 1.4166666666666667
41/29 - 1.4137931034482758
99/70 - 1.4142857142857144
239/169 - 1.4142011834319526
577/408 - 1.4142156862745099
1393/985 - 1.4142131979695431
3363/2378 - 1.4142136248948696
8119/5741 - 1.4142135516460548
19601/13860 - 1.4142135642135643
47321/33461 - 1.4142135620573204
114243/80782 - 1.4142135624272734
275807/195025 - 1.4142135623637995
665857/470832 - 1.4142135623746899
1607521/1136689 - 1.4142135623728214
3880899/2744210 - 1.4142135623731420
9369319/6625109 - 1.4142135623730870
22619537/15994428 - 1.4142135623730965

First 20 Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841
54971607658948646301386783144964782698772613513307493180078896702918825851648683235325858118170150873214978343601463118106546653220435805362395962991295556488036606954237309847762149971207793263738989
14030291214037674827921599320400561033992948898216351802670122530401263880575255235196727095109669287799074570417579539629351231775861429098849146880746524269269235328805333087546933690012894630670427794266440579064751300508834822795162874147983974059159392260220762973563561382652223360667198516093199367134903695783143116067743023134509886357032327271649
2434804314652199381956027075145741187716221548707931096877274520825143228915116227412484991366386864484767844200542482630246332092069382947111767723898168035847078557798454111405556629400142434835890123610082763986456199467423944182141028870863302603437534363208996458153115358483747994095302552907353919742211197822912892578751357668345638404394626711701120567186348490247426710813709165801137112237291901437566040249805155494297005186344325519103590369653438042689
346434895614929444828445967916634653215454504812454865104089892164276080684080254746939261017687341632569935171059945916359539268094914543114024020158787741692287531903178502306292484033576487391159597130834863729261484555671037916432206867189514675750227327687799973497042239286045783392065227614939379139866240959756584073664244580698830046194724340448293320938108876004367449471918175071251610962540447986139876845105399212429593945098472125140242905536711601925585608153109062121115635939797709
32074710952523740376423283403256578238321646122759160107427497117576305397686814013623874765833543023397971470911301264845142006214276865917420065183527313421909784286074786922242104480428021290764613639424408361555091057197776876849282654018358993099016644054242247557103410808928387071991436781136646322261169941417916607548507224950058710729258466238995253184617782314756913932650536663800753256087990078866003788647079369825102832504351225446531057648755795494571534144773842019836572551455718577614678081652481281009

Indices of first 20 Pell primes:
2 3 5 11 13 29 41 53 59 89 97 101 167 181 191 523 929 1217 1301 1361

First 20 Newman-Shank-Williams numbers:
1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607

First 20 near isosceles right triangles:
(3, 4, 5)
(20, 21, 29)
(119, 120, 169)
(696, 697, 985)
(4059, 4060, 5741)
(23660, 23661, 33461)
(137903, 137904, 195025)
(803760, 803761, 1136689)
(4684659, 4684660, 6625109)
(27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)
(927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)
(31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)
(1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)
(36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065)
(1235216565974040, 1235216565974041, 1746860020068409)```

## Phix

```with javascript_semantics
sequence p = {0,1},
pl = {2,2}
for i=2 to 41 do
p &= 2*p[i]+p[i-1]
pl &= 2*pl[i]+pl[i-1]
end for
printf(1,"First 20 Pell numbers: %s\n",{join_by(p[1..20],1,20," ",fmt:="%d")})
printf(1,"First 20 Pell-Lucas: %s\n",{join_by(pl[1..20],1,20," ",fmt:="%d")})
printf(1,"First 20 rational approximations of sqrt(2) (%.16f):\n",{sqrt(2)})
for i=2 to 21 do
integer n = pl[i]/2, d = p[i]
printf(1,"%d/%d ~= %.16g\n", {n,d,n/d})
end for
printf(1,"\nFirst 20 Pell primes:\n")
include mpfr.e
mpz {p0,p1,p2} = mpz_inits(3,{0,1,0})
sequence ppdx = {}
integer pdx = 2
while length(ppdx)<20 do
mpz_mul_si(p2,p1,2)
if is_prime(pdx) and mpz_prime(p2) then
printf(1,"%s\n",mpz_get_short_str(p2))
ppdx = append(ppdx,sprintf("%d",pdx))
end if
pdx += 1
mpz_set(p0,p1)
mpz_set(p1,p2)
end while
printf(1,"\nIndices of first 20 Pell primes: %s\n",join(ppdx," "))
sequence nsw = {}
for n=1 to 20 do nsw = append(nsw,sprintf("%d",p[2*n]+p[2*n-1])) end for
nsw[8..-3] = {"..."}
printf(1,"\nFirst 20 Newman-Shank-Williams numbers: %s\n",{join(nsw," ")})
printf(1,"\nFirst 20 near isosceles right triangles:\n")
for i=4 to 42 by 2 do
atom side = sum(p[1..i-1]), hypot = p[i]
printf(1,"[%d, %d, %d]\n", {side,side+1,hypot})
end for
```
Output:
```First 20 Pell numbers: 0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

First 20 Pell-Lucas: 2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

First 20 rational approximations of sqrt(2) (1.4142135623730951):
1/1 ~= 1
3/2 ~= 1.5
7/5 ~= 1.4
17/12 ~= 1.416666666666667
41/29 ~= 1.413793103448276
99/70 ~= 1.414285714285714
239/169 ~= 1.414201183431953
577/408 ~= 1.41421568627451
1393/985 ~= 1.414213197969543
3363/2378 ~= 1.41421362489487
8119/5741 ~= 1.414213551646055
19601/13860 ~= 1.414213564213564
47321/33461 ~= 1.41421356205732
114243/80782 ~= 1.414213562427273
275807/195025 ~= 1.414213562363799
665857/470832 ~= 1.41421356237469
1607521/1136689 ~= 1.414213562372821
3880899/2744210 ~= 1.414213562373142
9369319/6625109 ~= 1.414213562373087
22619537/15994428 ~= 1.414213562373097

First 20 Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
29647935552727996719...32712780937764687561 (64 digits)
67741382025708508432...87756429585398537901 (69 digits)
45562852543334487715...91808671945330235841 (73 digits)
54971607658948646301...49971207793263738989 (200 digits)
14030291214037674827...9886357032327271649 (356 digits)
24348043146521993819...3590369653438042689 (466 digits)
34643489561492944482...62121115635939797709 (498 digits)
32074710952523740376...14678081652481281009 (521 digits)

Indices of first 20 Pell primes: 2 3 5 11 13 29 41 53 59 89 97 101 167 181 191 523 929 1217 1301 1361

First 20 Newman-Shank-Williams numbers: 1 7 41 239 1393 8119 47321 ... 72722761475561 423859315570607

First 20 near isosceles right triangles:
[3, 4, 5]
[20, 21, 29]
[119, 120, 169]
[696, 697, 985]
[4059, 4060, 5741]
[23660, 23661, 33461]
[137903, 137904, 195025]
[803760, 803761, 1136689]
[4684659, 4684660, 6625109]
[27304196, 27304197, 38613965]
[159140519, 159140520, 225058681]
[927538920, 927538921, 1311738121]
[5406093003, 5406093004, 7645370045]
[31509019100, 31509019101, 44560482149]
[183648021599, 183648021600, 259717522849]
[1070379110496, 1070379110497, 1513744654945]
[6238626641379, 6238626641380, 8822750406821]
[36361380737780, 36361380737781, 51422757785981]
[211929657785303, 211929657785304, 299713796309065]
[1235216565974040, 1235216565974041, 1746860020068409]
```

## Raku

`my \$pell = cache lazy 0, 1, * + * × 2 … *;my \$pell-lucas = lazy 2, 2, * + * × 2 … *; my \$upto = 20; say   "First \$upto Pell numbers:\n" ~ \$pell[^\$upto]; say "\nFirst \$upto Pell-Lucas numbers:\n" ~ \$pell-lucas[^\$upto]; say "\nFirst \$upto rational approximations of √2 ({sqrt(2)}):\n" ~(1..\$upto).map({ sprintf "%d/%d - %1.16f", \$pell[\$_-1] + \$pell[\$_], \$pell[\$_], (\$pell[\$_-1]+\$pell[\$_])/\$pell[\$_] }).join: "\n"; say "\nFirst \$upto Pell primes:\n" ~ \$pell.grep(&is-prime)[^\$upto].join: "\n"; say "\nIndices of first \$upto Pell primes:\n" ~ (^∞).grep({\$pell[\$_].is-prime})[^\$upto]; say "\nFirst \$upto Newman-Shank-Williams numbers:\n" ~ (^\$upto).map({ \$pell[2 × \$_, 2 × \$_+1].sum }); say "\nFirst \$upto near isosceles right triangles:";map -> \p { printf "(%d, %d, %d)\n", |(\$_, \$_+1 given \$pell[^(2 × p + 1)].sum), \$pell[2 × p + 1] }, 1..\$upto;`
Output:
```First 20 Pell numbers:
0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

First 20 Pell-Lucas numbers:
2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

First 20 rational approximations of √2 (1.4142135623730951):
1/1 - 1.0000000000000000
3/2 - 1.5000000000000000
7/5 - 1.4000000000000000
17/12 - 1.4166666666666667
41/29 - 1.4137931034482758
99/70 - 1.4142857142857144
239/169 - 1.4142011834319526
577/408 - 1.4142156862745099
1393/985 - 1.4142131979695431
3363/2378 - 1.4142136248948696
8119/5741 - 1.4142135516460548
19601/13860 - 1.4142135642135643
47321/33461 - 1.4142135620573204
114243/80782 - 1.4142135624272734
275807/195025 - 1.4142135623637995
665857/470832 - 1.4142135623746899
1607521/1136689 - 1.4142135623728214
3880899/2744210 - 1.4142135623731420
9369319/6625109 - 1.4142135623730870
22619537/15994428 - 1.4142135623730965

First 20 Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841
54971607658948646301386783144964782698772613513307493180078896702918825851648683235325858118170150873214978343601463118106546653220435805362395962991295556488036606954237309847762149971207793263738989
14030291214037674827921599320400561033992948898216351802670122530401263880575255235196727095109669287799074570417579539629351231775861429098849146880746524269269235328805333087546933690012894630670427794266440579064751300508834822795162874147983974059159392260220762973563561382652223360667198516093199367134903695783143116067743023134509886357032327271649
2434804314652199381956027075145741187716221548707931096877274520825143228915116227412484991366386864484767844200542482630246332092069382947111767723898168035847078557798454111405556629400142434835890123610082763986456199467423944182141028870863302603437534363208996458153115358483747994095302552907353919742211197822912892578751357668345638404394626711701120567186348490247426710813709165801137112237291901437566040249805155494297005186344325519103590369653438042689
346434895614929444828445967916634653215454504812454865104089892164276080684080254746939261017687341632569935171059945916359539268094914543114024020158787741692287531903178502306292484033576487391159597130834863729261484555671037916432206867189514675750227327687799973497042239286045783392065227614939379139866240959756584073664244580698830046194724340448293320938108876004367449471918175071251610962540447986139876845105399212429593945098472125140242905536711601925585608153109062121115635939797709
32074710952523740376423283403256578238321646122759160107427497117576305397686814013623874765833543023397971470911301264845142006214276865917420065183527313421909784286074786922242104480428021290764613639424408361555091057197776876849282654018358993099016644054242247557103410808928387071991436781136646322261169941417916607548507224950058710729258466238995253184617782314756913932650536663800753256087990078866003788647079369825102832504351225446531057648755795494571534144773842019836572551455718577614678081652481281009

Indices of first 20 Pell primes:
2 3 5 11 13 29 41 53 59 89 97 101 167 181 191 523 929 1217 1301 1361

First 20 Newman-Shank-Williams numbers:
1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607

First 20 near isosceles right triangles:
(3, 4, 5)
(20, 21, 29)
(119, 120, 169)
(696, 697, 985)
(4059, 4060, 5741)
(23660, 23661, 33461)
(137903, 137904, 195025)
(803760, 803761, 1136689)
(4684659, 4684660, 6625109)
(27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)
(927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)
(31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)
(1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)
(36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065)
(1235216565974040, 1235216565974041, 1746860020068409)```

## Scala

`val pellNumbers: LazyList[BigInt] =  BigInt("0") #:: BigInt("1") #::    (pellNumbers zip pellNumbers.tail).      map{ case (a,b) => 2*b + a } val pellLucasNumbers: LazyList[BigInt] =  BigInt("2") #:: BigInt("2") #::    (pellLucasNumbers zip pellLucasNumbers.tail).      map{ case (a,b) => 2*b + a } val pellPrimes: LazyList[BigInt] =  pellNumbers.tail.tail.    filter{ case p => p.isProbablePrime(16) } val pellIndexOfPrimes: LazyList[BigInt] =  pellNumbers.tail.tail.    filter{ case p => p.isProbablePrime(16) }.    map{ case p => pellNumbers.indexOf(p) } val pellNSWnumbers: LazyList[BigInt] =  (pellNumbers.zipWithIndex.collect{ case (p,i) if i%2 == 0 => p}    zip   pellNumbers.zipWithIndex.collect{ case (p,i) if i%2 != 0 => p}).    map{ case (a,b) => a + b } val pellSqrt2Numerator: LazyList[BigInt] =  BigInt(1) #:: BigInt(3) #::    (pellSqrt2Numerator zip pellSqrt2Numerator.tail).      map{ case (a,b) => 2*b + a } val pellSqrt2: LazyList[BigDecimal] =  (pellSqrt2Numerator zip pellNumbers.tail).    map{ case (n,d) => BigDecimal(n)/BigDecimal(d) } val pellSqrt2asString: LazyList[String] =  (pellSqrt2Numerator zip pellNumbers.tail).    map{ case (n,d) => s"\$n/\$d" } val pellHypotenuse: LazyList[BigInt] =  pellNumbers.tail.tail.zipWithIndex.collect{ case (p,i) if i%2 != 0 => p } val pellShortLeg: LazyList[BigInt] =  LazyList.from(3,2).map{ case s => pellNumbers.take(s).sum } val pellTriple: LazyList[(BigInt,BigInt,BigInt)] =  (pellHypotenuse zip pellShortLeg).    map{ case (h,s) => (s,s+1,h)} // Output{ println("7 Tasks")  println("-------")  println(pellNumbers.take(10).mkString("1. Pell Numbers: ", ",", "\n"))  println(pellLucasNumbers.take(10).mkString("2. Pell-Lucas Numbers: ", ",", "\n"))  println((pellSqrt2asString zip pellSqrt2).take(10).    map { case (f, d) => s"\$f = \$d" }.    mkString("3. Square-root of 2 Approximations: \n\n",      "\n", "\n"))  println(pellPrimes.take(10).mkString("4. Pell Primes: \n\n", "\n", "\n"))  println(pellIndexOfPrimes.take(10).mkString("5. Pell Index of Primes: ", ",", "\n"))  println(pellNSWnumbers.take(10).mkString("6. Newman-Shank-Williams Numbers: \n\n", "\n", "\n"))  println(pellTriple.take(10).mkString("7. Near Right-triangle Triples: \n\n", "\n", "\n"))} `
Output:
```7 Tasks
-------
1. Pell Numbers: 0,1,2,5,12,29,70,169,408,985

2. Pell-Lucas Numbers: 2,2,6,14,34,82,198,478,1154,2786

3. Square-root of 2 Approximations:

1/1 = 1
3/2 = 1.5
7/5 = 1.4
17/12 = 1.416666666666666666666666666666667
41/29 = 1.413793103448275862068965517241379
99/70 = 1.414285714285714285714285714285714
239/169 = 1.414201183431952662721893491124260
577/408 = 1.414215686274509803921568627450980
1393/985 = 1.414213197969543147208121827411168
3363/2378 = 1.414213624894869638351555929352397

4. Pell Primes:

2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449

5. Pell Index of Primes: 2,3,5,11,13,29,41,53,59,89

6. Newman-Shank-Williams Numbers:

1
7
41
239
1393
8119
47321
275807
1607521
9369319

7. Near Right-triangle Triples:

(3,4,5)
(20,21,29)
(119,120,169)
(696,697,985)
(4059,4060,5741)
(23660,23661,33461)
(137903,137904,195025)
(803760,803761,1136689)
(4684659,4684660,6625109)
(27304196,27304197,38613965)```

## Wren

Library: Wren-big
Library: Wren-math
Library: Wren-fmt
`import "./big" for BigInt, BigRatimport "./math" for Intimport "./fmt" for Fmt var p = List.filled(40, 0)p[0] = 0p[1] = 1for (i in 2..39) p[i] = 2 * p[i-1] + p[i-2]System.print("The first 20 Pell numbers are:")System.print(p[0..19].join(" ")) var q = List.filled(40, 0)q[0] = 2q[1] = 2for (i in 2..39) q[i] = 2 * q[i-1] + q[i-2]System.print("\nThe first 20 Pell-Lucas numbers are:")System.print(q[0..19].join(" ")) System.print("\nThe first 20 rational approximations of √2 (1.4142135623730951) are:")for (i in 1..20) {    var r = BigRat.new(q[i]/2, p[i])    Fmt.print("\$-17s ≈ \$-18s", r, r.toDecimal(16, true, true))} System.print("\nThe first 15 Pell primes are:")var p0 = BigInt.zerovar p1 = BigInt.onevar indices = List.filled(15, 0)var count = 0var index = 2var p2while (count < 15) {    p2 = p1 * BigInt.two + p0    if (Int.isPrime(index) && p2.isProbablePrime(10)) {        System.print(p2)        indices[count] = index        count = count + 1    }    index = index + 1    p0 = p1    p1 = p2} System.print("\nIndices of the first 15 Pell primes are:")System.print(indices.join(" ")) System.print("\nFirst 20 Newman-Shank_Williams numbers:")var nsw = List.filled(20, 0)for (n in 0..19) nsw[n] = p[2*n] + p[2*n+1]Fmt.print("\$d", nsw) System.print("\nFirst 20 near isosceles right triangles:")p0 = 0p1 = 1var sum = 1var i = 2while (i < 43) {    p2 = p1 * 2 + p0    if (i % 2 == 1) {        Fmt.print("(\$d, \$d, \$d)", sum, sum + 1, p2)    }    sum = sum + p2    p0 = p1    p1 = p2    i = i + 1}`
Output:
```The first 20 Pell numbers are:
0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

The first 20 Pell-Lucas numbers are:
2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

The first 20 rational approximations of √2 (1.4142135623730951) are:
1/1               ≈ 1.0000000000000000
3/2               ≈ 1.5000000000000000
7/5               ≈ 1.4000000000000000
17/12             ≈ 1.4166666666666667
41/29             ≈ 1.4137931034482759
99/70             ≈ 1.4142857142857143
239/169           ≈ 1.4142011834319527
577/408           ≈ 1.4142156862745098
1393/985          ≈ 1.4142131979695431
3363/2378         ≈ 1.4142136248948696
8119/5741         ≈ 1.4142135516460547
19601/13860       ≈ 1.4142135642135642
47321/33461       ≈ 1.4142135620573205
114243/80782      ≈ 1.4142135624272734
275807/195025     ≈ 1.4142135623637995
665857/470832     ≈ 1.4142135623746899
1607521/1136689   ≈ 1.4142135623728214
3880899/2744210   ≈ 1.4142135623731420
9369319/6625109   ≈ 1.4142135623730870
22619537/15994428 ≈ 1.4142135623730964

The first 15 Pell primes are:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841

Indices of the first 15 Pell primes are:
2 3 5 11 13 29 41 53 59 89 97 101 167 181 191

First 20 Newman-Shank_Williams numbers:
1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607

First 20 near isosceles right triangles:
(3, 4, 5)
(20, 21, 29)
(119, 120, 169)
(696, 697, 985)
(4059, 4060, 5741)
(23660, 23661, 33461)
(137903, 137904, 195025)
(803760, 803761, 1136689)
(4684659, 4684660, 6625109)
(27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)
(927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)
(31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)
(1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)
(36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065)
(1235216565974040, 1235216565974041, 1746860020068409)
```