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.

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

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.


Task
  • 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.


See also



ALGOL 68

Translation of: FreeBASIC – using Miller Rabin for finding Pell primes and some minor output format differences and showing 10 Pell primes
Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Should work with any implementation of Algol 68 if LONG INT is large enough to hold the 90th Pell number/

BEGIN # find some Pell numbers - trans FreeBASIC ( which is trans Phix )     #

    PR read "primes.incl.a68" PR

    [ 0 : 90 ]LONG INT p, pl;
    p[  0 ] := 0; p[  1 ] := 1;
    pl[ 0 ] := 2; pl[ 1 ] := 2;
    FOR n FROM 2 TO UPB p DO
        p[  n ] := 2 * p[  n - 1 ] + p[  n - 2 ];
        pl[ n ] := 2 * pl[ n - 1 ] + pl[ n - 2 ]
    OD;

    print( ( "First 20 Pell numbers:", newline ) );
    FOR n FROM 0 TO 19 DO print( ( " ", whole( p[  n ], 0 ) ) ) OD;
    print( ( newline, newline, "First 20 Pell-Lucas numbers:", newline ) );
    FOR n FROM 0 TO 19 DO print( ( " ", whole( pl[ n ], 0 ) ) ) OD;

    print( ( newline, newline, "First 20 rational approximations of sqrt(2) (" ) );
    print( ( fixed( sqrt( 2 ), -15, 13 ), "):", newline ) );
    FOR n TO 20 DO
        LONG INT j = pl[ n ] OVER 2, d = p[ n ];
        print( ( " ", whole( j, 0 ), "/", whole( d, 0 ), " ~= ", fixed( j / d, -15, 13 ), newline ) )
    OD;

    print( ( newline, "First 10 Pell primes:", newline, "index Pell prime", newline ) );
    INT c := 0;
    FOR pdx FROM 2 WHILE c < 10 DO
        IF is probably prime( p[ pdx ] ) THEN 
            print( ( whole( pdx, -5 ), " ", whole( p[ pdx ], 0 ), newline ) );
            c +:= 1 
        FI
    OD;

    print( ( newline, newline, "First 20 Newman-Shank-Williams numbers:", newline ) );
    FOR n FROM 0 TO 19 DO
        LONG INT nsw = p[ 2 * n ] + p[ 2 * n + 1 ];
        print( ( " ", whole( nsw, 0 ) ) ); IF n = 13 THEN print( ( newline ) ) FI
    OD;

    print( ( newline, newline, "First 20 near isosceles right triangles:", newline ) );
    LONG INT i0 := 0, i1 := 1, t := 1, found := 0;
    FOR i FROM 2 WHILE found < 20 DO
        LONG INT i2 = i1*2 + i0;
        IF ODD i THEN
            print( ( " [", whole( t, 0 ), ", ", whole( t + 1, 0 ), ", ", whole( i2, 0 ), "]", newline ) );
            found +:= 1
        FI;
        t +:= i2; i0 := i1; i1 := i2
    OD

END
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 sqrt(2) (1.4142135623731):
 1/1 ~= 1.0000000000000
 3/2 ~= 1.5000000000000
 7/5 ~= 1.4000000000000
 17/12 ~= 1.4166666666667
 41/29 ~= 1.4137931034483
 99/70 ~= 1.4142857142857
 239/169 ~= 1.4142011834320
 577/408 ~= 1.4142156862745
 1393/985 ~= 1.4142131979695
 3363/2378 ~= 1.4142136248949
 8119/5741 ~= 1.4142135516461
 19601/13860 ~= 1.4142135642136
 47321/33461 ~= 1.4142135620573
 114243/80782 ~= 1.4142135624273
 275807/195025 ~= 1.4142135623638
 665857/470832 ~= 1.4142135623747
 1607521/1136689 ~= 1.4142135623728
 3880899/2744210 ~= 1.4142135623731
 9369319/6625109 ~= 1.4142135623731
 22619537/15994428 ~= 1.4142135623731

First 10 Pell primes:
index Pell prime
    2 2
    3 5
    5 29
   11 5741
   13 33461
   29 44560482149
   41 1746860020068409
   53 68480406462161287469
   59 13558774610046711780701
   89 4125636888562548868221559797461449


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]

ALGOL W

Translation of: FreeBASIC – via Algol 68 - basic task only and only 5 Pell primes as Algol W limits integers to 32-bit.
begin % find some Pell numbers - trans FreeBasic ( which is trans Phix )     %

    % returns true if n is prime, false otherwise, uses trial division       %
    logical procedure isPrime ( integer value n ) ;
        if      n < 3        then n = 2
        else if n rem 3 = 0  then n = 3
        else if not odd( n ) then false
        else begin
            logical prime;
            integer f, f2, toNext;
            prime  := true;
            f      := 5;
            f2     := 25;
            toNext := 24;           % note: ( 2n + 1 )^2 - ( 2n - 1 )^2 = 8n %
            while f2 <= n and prime do begin
                prime  := n rem f not = 0;
                f      := f + 2;
                f2     := toNext;
                toNext := toNext + 8
             end while_f2_le_n_and_prime ;
             prime
        end isPrime ;

    integer MAX_P;
    MAX_P := 9;
    begin
        integer array p, pl ( 0 :: 20 );    % need more than 10 Pell numbers %
        integer c, pdx;                       % to find the fifth Pell prime %

        p(  0 ) := 0; p(  1 ) := 1;
        pl( 0 ) := 2; pl( 1 ) := 2;
        for n := 2 until 20 do begin
            p(  n ) := 2 * p(  n - 1 ) + p(  n - 2 );
            pl( n ) := 2 * pl( n - 1 ) + pl( n - 2 )
        end for_n ;

        write( "First 10 Pell numbers:" );
        for n := 0 until MAX_P do begin writeon( i_w := 1, s_w := 0, " ", p( n ) ) end;
        write();write( "First 10 Pell-Lucas numbers:" );
        for n := 0 until MAX_P do begin
            writeon( i_w := 1, s_w := 0, " ", pl( n ) )
        end for_n ;

        write( s_w := 0, "First 10 rational approximations of sqrt(2) (" );
        writeon( s_w := 0, r_format := "A", r_w := 8, r_d := 6, sqrt( 2 ), "):" );
        for n := 1 until MAX_P do begin
            integer j;
            j := pl( n ) div 2;
            write( i_w := 1, s_w := 0, " ", j, "/", p( n ), " ~= "
                 , r_format := "A", r_w := 8, r_d := 6, j / p( n )
                 )
        end for_n;

        write();write( "First 5 Pell primes:" ); write( "index Pell prime" );
        c := 0;
        pdx := 2;
        while c < 5 do begin
            if isPrime( p( pdx ) ) then begin 
                write( i_w := 5, s_w := 0, pdx, " ", i_w := 1, p( pdx ) );
                c := c + 1 
            end if_isPrime_p_pdx ;
            pdx := pdx + 1
        end while_c_lt_5 ;

        write(); write( "First 10 Newman-Shank-Williams numbers:" );write();
        for n := 0 until MAX_P do begin
            integer nsw;
            nsw := p( 2 * n ) + p( 2 * n + 1 );
            writeon( i_w := 1, s_w := 0, " ", nsw )
        end for_n;

        write();write( "First 10 near isosceles right triangles:" );
        begin
            integer i, i0, i1, i2, t, found;
            i0 := 0; i1 := 1; t := 1; found := 0;
            i := 1;
            while found < 10 do begin
                i  := i + 1;
                i2 := i1*2 + i0;
                if odd( i ) then begin
                    write( i_w := 1, s_w := 0, " [", t, ", ", t + 1, ", ", i2, "]" );
                    found := found + 1
                end if_odd_i ;
                t := t + i2; i0 := i1; i1 := i2
            end while_found_lt_10
        end

    end

end.
Output:
First 10 Pell numbers: 0 1 2 5 12 29 70 169 408 985
First 10 Pell-Lucas numbers: 2 2 6 14 34 82 198 478 1154 2786

First 10 rational approximations of sqrt(2) (1.414213):
 1/1 ~= 1.000000
 3/2 ~= 1.500000
 7/5 ~= 1.400000
 17/12 ~= 1.416666
 41/29 ~= 1.413793
 99/70 ~= 1.414285
 239/169 ~= 1.414201
 577/408 ~= 1.414215
 1393/985 ~= 1.414213

First 5 Pell primes:
index Pell prime
    2 2
    3 5
    5 29
   11 5741
   13 33461

First 10 Newman-Shank-Williams numbers:
 1 7 41 239 1393 8119 47321 275807 1607521 9369319

First 10 near isosceles right triangles:
 [2, 3, 5]
 [12, 13, 29]
 [70, 71, 169]
 [408, 409, 985]
 [2378, 2379, 5741]
 [13860, 13861, 33461]
 [80782, 80783, 195025]
 [470832, 470833, 1136689]
 [2744210, 2744211, 6625109]
 [15994428, 15994429, 38613965]

Common Lisp

(defun recurrent-sequence-2 (a0 a1 k1 k2 max)
 "A generic function for any recurrent sequence of order 2, where a0 and a1 are the initial elements, 
  k1 is the factor of a(n-1) and k2 is the factor of a(n-2)"
	(do* ((i 0 (1+ i))
	      (result (list a1 a0))
	      (b0 a0 b1)
	      (b1 a1 b2)
	      (b2 (+ (* k1 b1) (* k2 b0)) (+ (* k1 b1) (* k2 b0))) )
	  	((> i max) (nreverse result))
		(push b2 result) ))

(defun pell-sequence (max)
	(recurrent-sequence-2 0 1 2 1 max) )

(defun pell-lucas-sequence (max)
	(recurrent-sequence-2 2 2 2 1 max) )

(defun fibonacci-sequence (max) ; As an extra bonus, you get Fibonacci's numbers with this simple call
	(recurrent-sequence-2 1 1 1 1 max) )

(defun rational-approximation-sqrt2 (max)
 "Approximate square root of 2 with (P(n-1)+P(n-2))/P(n)"
	(butlast (maplist #'(lambda (l) (/ (+ (first l) (or (second l) 0)) (or (second l) 1))) (pell-sequence max))) )

(defun pell-primes (max)
	(do* ((i 0 (1+ i))
	      (result (list 1 0))
	      (indices nil)
	      (b0 0 b1)
	      (b1 1 b2)
	      (b2 (+ (* 2 b1) (* 1 b0)) (+ (* 2 b1) (* 1 b0))) )
	  	((> (length result) max) (values (nreverse result)(nreverse indices)))
      ; primep can be any function determining whether a number is prime, for example, 
      ; https://rosettacode.org/wiki/Primality_by_Wilson%27s_theorem#Common_Lisp
	  (when (primep b2)
			(push b2 result)
			(push i indices) )))


The first 10 Pell numbers are:

(pell-sequence 10)
(0 1 2 5 12 29 70 169 408 985 2378 5741 13860)

The first 10 Pell-Lucas numbers are:

(pell-lucas-sequence 10)
(2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202)

Rational approximation of square root of 2:

(rational-approximation-sqrt2 10)
(1 3/2 7/5 17/12 41/29 99/70 239/169 577/408 1393/985 3363/2378 8119/5741
 19601/13860)

The same in decimal form:

(mapcar #'float (rational-approximation-sqrt2 10))
(1.0 1.5 1.4 1.4166666 1.4137931 1.4142857 1.4142011 1.4142157 1.4142132
 1.4142137 1.4142135 1.4142135)

First 7 Pell primes and, as second value, the corresponding indices:

(pell-primes 7)
(0 1 2 5 29 5741 33461 44560482149) ;
(0 1 3 9 11 27)

Still missing some of the taks

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 True
End Function

Dim As Integer n
Dim As Integer p(0 To 40), pl(0 To 40)
p(0)= 0: p(1) = 1
pl(0) = 2: pl(1) = 2
For 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 n
Print !"\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/d
Next n

Print !"\nFirst 6 Pell primes: [for the limitations of the FB standard library]"
Dim as Integer pdx = 2
Dim As Byte c = 0
Dim 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 += 1
loop 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 n
Print !"\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 = 0
Do 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 += 1
Loop
Sleep
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)

Haskell

import Data.Numbers.Primes (isPrime)

----------------------- PELL SERIES ----------------------

pell :: Integer -> Integer -> [Integer]
pell a b = a : b : zipWith (+) (pell a b) ((2 *) <$> tail (pell a b))

a000129, a002203, a001333, a086383, a096650, a002315 :: [Integer]
a000129 = pell 0 1

a002203 = pell 2 2

a001333 = (`div` 2) <$> a002203

a086383 = filter isPrime a000129

a096650 = zip [0 ..] a000129 >>= (\(i, n) -> [i | isPrime n])

a002315 = 1 : 7 : zipWith (-) ((6 *) <$> tail a002315) a002315


------------------- PYTHAGOREAN TRIPLES ------------------

pythagoreanTriples :: [(Integer, Integer, Integer)]
pythagoreanTriples =
  (tail . concat) $ zipWith3 go [0 ..] a000129 (scanl (+) 0 a000129)
  where
    go i p m
      | odd i = [(m, succ m, p)]
      | otherwise = []

-------------------------- TESTS -------------------------
main :: IO ()
main = do
  mapM_
    (\(k, xs) -> putStrLn ('\n' : k) >> print (take 10 xs))
    [ ("a000129", a000129)
    , ("a002203", a002203)
    , ("a001333", a001333)
      -- Waste of electrical power ?
      -- ("a086383", a086383)
      -- ("a096650", a096650
    , ("a002315", a002315)
    ]
    
  putStrLn "\nRational approximations to sqrt 2:"
  mapM_ putStrLn $
    (take 10 . tail) $
    zipWith
      (\n d ->
         show n <>
         ('/' : show d) <> " -> " <> show (fromIntegral n / fromIntegral d))
      a001333
      a000129
     
  putStrLn "\nPythagorean triples:"
  mapM_ print $ take 10 pythagoreanTriples
Output:
a000129
[0,1,2,5,12,29,70,169,408,985]

a002203
[2,2,6,14,34,82,198,478,1154,2786]

a001333
[1,1,3,7,17,41,99,239,577,1393]

a002315
[1,7,41,239,1393,8119,47321,275807,1607521,9369319]

Rational approximations to sqrt 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

Pythagorean 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)

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 extender
Pn=: (%:8) %~(1+%:2)&^ - (1-%:2)&^ NB. pell, closed form
Qn=: (1+%:2)&^ + (1-%:2)&^         NB. pell lucas, closed form
QN=: +: %&Pn ]                     NB. pell lucas, closed form
qn=: 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.11
0 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.11
2 2 6 14 34 82 198 478 1154 2786 6726
   QN i.11
0 2 6 14 34 82 198 478 1154 2786 6726
   qn i.11
2 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.11
2 2 6 14 34 82 198 478 1154 2786 6726

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

   }.(%~ _1}. +//.@,:~) nextPell^:9(0 1)
1 1.5 1.4 1.41667 1.41379 1.41429 1.4142 1.41422 1.41421 1.41421
   }.(%~ _1}. +//.@,:~) 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  6625109
27304196 27304197 38613965

Java

import java.math.BigDecimal;
import java.math.BigInteger;
import java.math.MathContext;
import java.util.ArrayList;
import java.util.List;

public final class PellNumbers {

	public static void main(String[] aArgs) {		
		System.out.println("7 Tasks");
		System.out.println("-------");
		System.out.println("Task 1, Pell Numbers: " + pellNumbers(TERM_COUNT));
		System.out.println();
		System.out.println("Task 2, Pell-Lucas Numbers: " + pellLucasNumbers(TERM_COUNT));
		System.out.println();
		System.out.println("Task 3, Approximations to square root of 2:");
		List<Rational> rationals = squareRoot2(TERM_COUNT + 1);
		for ( int i = 1; i < TERM_COUNT + 1; i++ ) {
			System.out.println(rationals.get(i) + " = " + rationals.get(i).toBigDecimal());
		}
		System.out.println();
		List<Pair> pairs = pellPrimes(TERM_COUNT);
		System.out.println("Task 4, Pell primes:");
		for ( int i = 0; i < TERM_COUNT; i++ ) {
			System.out.println(pairs.get(i).pellPrime);
		}
		System.out.println();
		System.out.print("Task 5, Pell indices of Pell primes:");
		for ( int i = 0; i < TERM_COUNT; i++ ) {
			System.out.print(pairs.get(i).index + " ");
		}
		System.out.println();
		System.out.println();
		System.out.println("Task 6, Newman-Shank-Williams numbers: " + newmanShankWilliams(TERM_COUNT));
		System.out.println();		
		System.out.println("Task , Near isoseles right triangles:");
		List<Triple> nearIsoselesRightTriangles = nearIsoslesRightTriangles(TERM_COUNT);
		for ( int i = 0; i < TERM_COUNT; i++ ) {
			System.out.println(nearIsoselesRightTriangles.get(i));
		}		
	}
	
	private static List<BigInteger> pellNumbers(int aTermCount) {
		PellIterator pellIterator = new PellIterator(BigInteger.ZERO, BigInteger.ONE);
		List<BigInteger> result = new ArrayList<BigInteger>();
		for ( int i = 0; i < aTermCount; i++ ) {			
			result.add(pellIterator.next());
		}
		return result;
	}
	
	private static List<BigInteger> pellLucasNumbers(int aTermCount) {
		PellIterator pellLucasIterator = new PellIterator(BigInteger.TWO, BigInteger.TWO);
		List<BigInteger> result = new ArrayList<BigInteger>();
		for ( int i = 0; i < aTermCount; i++ ) {
			result.add(pellLucasIterator.next());
		}
		return result;
	}
	
	private static List<Rational> squareRoot2(int aTermCount) {
		PellIterator pellIterator = new PellIterator(BigInteger.ZERO, BigInteger.ONE);
		PellIterator pellLucasIterator = new PellIterator(BigInteger.TWO, BigInteger.TWO);
		List<Rational> result = new ArrayList<Rational>();
		for ( int i = 0; i < aTermCount; i++ ) {
			result.add( new Rational(pellLucasIterator.next().divide(BigInteger.TWO), pellIterator.next()) );
		}
		return result;		
	}
	
	private static List<Pair> pellPrimes(int aTermCount) {
		PellIterator pellIterator = new PellIterator(BigInteger.ZERO, BigInteger.ONE);
		int index = 0;
		int count = 0;
		List<Pair> result = new ArrayList<Pair>();
		while ( count < aTermCount ) {
			BigInteger pellNumber = pellIterator.next();
			if ( pellNumber.isProbablePrime(16) ) {
				result.add( new Pair(pellNumber, index) );
				count += 1;
			}
			index += 1;		
		}
		return result;
	}
	
	private static List<BigInteger> newmanShankWilliams(int aTermCount) {
		PellIterator pellIterator = new PellIterator(BigInteger.ZERO, BigInteger.ONE);
		List<BigInteger> result = new ArrayList<BigInteger>();
		for ( int i = 0; i < aTermCount; i++ ) {
			BigInteger pellNumber = pellIterator.next();
			result.add(pellNumber.add(pellIterator.next()));
		}		
		return result;
	}
	
	private static List<Triple> nearIsoslesRightTriangles(int aTermCount) {
		PellIterator pellIterator = new PellIterator(BigInteger.ZERO, BigInteger.ONE);
		pellIterator.next();
		List<Triple> result = new ArrayList<Triple>();
		BigInteger sum = pellIterator.next();
		for ( int i = 0; i < aTermCount; i++ ) {
			sum = sum.add(pellIterator.next());
			BigInteger nextTerm = pellIterator.next();
			result.add( new Triple(sum, sum.add(BigInteger.ONE), nextTerm) );
			sum = sum.add(nextTerm);
		}
		return result;
	}
		
	private static class PellIterator {
			
		public PellIterator(BigInteger aFirst, BigInteger aSecond) {
			a = aFirst; b = aSecond;
		}
		
		public BigInteger next() {
			aCopy = a;
			bCopy = b;
			b = b.add(b).add(a);
			a = bCopy;
			return aCopy;
		}
		
		private BigInteger a, aCopy, b, bCopy;
		
	}
	
	private static record Rational(BigInteger numerator, BigInteger denominator) {
		
		public BigDecimal toBigDecimal() {
			return new BigDecimal(numerator).divide( new BigDecimal(denominator), mathContext );
		}
		
		public String toString() {
			return numerator + " / " + denominator;
		}
		
		private static MathContext mathContext = new MathContext(34);
		
	}
	
	private static record Pair(BigInteger pellPrime, int index) {}
	
	private static record Triple(BigInteger shortSide, BigInteger longSide, BigInteger hypotenuse) {
		
		public String toString() {
			return "(" + shortSide + ", " + longSide + ", " + hypotenuse + ")" ;
		}
		
	}
	
	private static final int TERM_COUNT = 10;
	
}
Output:
7 Tasks
-------
Task 1, Pell Numbers: [0, 1, 2, 5, 12, 29, 70, 169, 408, 985]

Task 2, Pell-Lucas Numbers: [2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786]

Task 3, Approximations to square root of 2:
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

Task 4, Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449

Task 5, Pell indices of Pell primes:2 3 5 11 13 29 41 53 59 89 

Task 6, Newman-Shank-Williams numbers: [1, 7, 41, 239, 1393, 8119, 47321, 275807, 1607521, 9369319]

Task , Near isoseles 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)

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 pells
end

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 pelllucas
end

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, pellprimes
end

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)

Mathematica/Wolfram Language

ClearAll[PellNumber, PellLucasNumber]
PellNumber[0] = 0;
PellNumber[1] = 1;
PellNumber[n_] := PellNumber[n] = 2 PellNumber[n - 1] + PellNumber[n - 2]

PellLucasNumber[0] = 2;
PellLucasNumber[1] = 2;
PellLucasNumber[n_] := PellLucasNumber[n] = 2 PellLucasNumber[n - 1] + PellLucasNumber[n - 2]

pns = PellNumber /@ Range[0, 9]

plns = PellLucasNumber /@ Range[0, 9]

den = Rest@pns;
num = Rest@plns/2;
approx = num/den
N[approx]

pns = {#, PellNumber[#]} & /@ Range[0, 100];
Select[pns, Last/*PrimeQ, 10] // Grid

ClearAll[PellS]
PellS[n_] := If[n == 0, 1, PellNumber[2 n] + PellNumber[2 n + 1]]
PellS /@ Range[0, 19]

ClearAll[PythagoreanTriple]
PythagoreanTriple[n_Integer] := Module[{hypo, short, long},
  hypo = PellNumber[2 n + 1];
  short = Total[PellNumber /@ Range[2 n]];
  long = short + 1;
  {short, long, hypo}
 ]
PythagoreanTriple /@ Range[10]
Output:
{0, 1, 2, 5, 12, 29, 70, 169, 408, 985}

{2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786}

{1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985}

{1., 1.5, 1.4, 1.41667, 1.41379, 1.41429, 1.4142, 1.41422, 1.41421}

2	2
3	5
5	29
11	5741
13	33461
29	44560482149
41	1746860020068409
53	68480406462161287469
59	13558774610046711780701
89	4125636888562548868221559797461449

{1, 7, 41, 239, 1393, 8119, 47321, 275807, 1607521, 9369319, 54608393, 318281039, 1855077841, 10812186007, 63018038201, 367296043199, 2140758220993, 12477253282759, 72722761475561, 423859315570607}

{{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}}

Nim

import std/[strformat, strutils, sugar]
import integers

template isOdd(n: int): bool = (n and 1) != 0

iterator pellSequence[T](first, second: T; lim = -1): (int, T) =
  ## Yield the sucessive values of a Pell or Pell-Lucas sequence
  ## preceded by their rank.
  ## If "lim" is specified and greater than 2, only the "lim" first
  ## values are computed.
  ## The iterator works with "int" values or "Integer" values.
  var prev = first
  var curr = second
  yield (0, prev)
  yield (1, curr)
  var count = 2
  while true:
    swap prev, curr
    curr += 2 * prev
    yield (count, curr)
    inc count
    if count == lim: break

echo "First 10 Pell numbers:"
let p = collect:
          for (idx, val) in pellSequence(0, 1, 11): val
echo p[0..9].join(" ")

echo "\nFirst 10 Pell-Lucas numbers:"
let q = collect:
          for (idx, val) in pellSequence(2, 2, 11): val
echo q[0..9].join(" ")

echo "\nFirst 10 rational approximations of √2:"
for i in 1..10:
  let n = q[i] div 2
  let d = p[i]
  let r = &"{n}/{d}"
  echo &"{r:>9} = {n/d:.17}"

echo "\nFirst 10 Pell primes:"
# To avoid an overflow, we need to use Integer values here.
var indices: seq[int]
var count = 0
for (idx, p) in pellSequence(newInteger(0), newInteger(1)):
  if p.isPrime:
    echo p
    indices.add idx
    inc count
    if count == 10: break

echo "\nFirst 10 Pell primes indices:"
echo indices.join(" ")

echo "\nFirst 10 Newman-Shank-Williams numbers:"
count = 0
var prev: int
for (idx, p) in pellSequence(0, 1):
  if idx.isOdd:
    inc count
    stdout.write prev + p
    if count == 10: break
    stdout.write ' '
  else:
    prev = p
echo()

echo "\nFirst 10 near isosceles right triangles:"
count = 0
var sum = 0
for (idx, p) in pellSequence(0, 1):
  if idx.isOdd and sum != 0:
    echo (sum, sum + 1, p)
    inc count
    if count == 10: break
  inc sum, p
Output:
First 10 Pell numbers:
0 1 2 5 12 29 70 169 408 985

First 10 Pell-Lucas numbers:
2 2 6 14 34 82 198 478 1154 2786

First 10 rational approximations of √2:
      1/1 = 1.0000000000000000
      3/2 = 1.5000000000000000
      7/5 = 1.3999999999999999
    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

First 10 Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449

First 10 Pell primes indices:
2 3 5 11 13 29 41 53 59 89

First 10 Newman-Shank-Williams numbers:
1 7 41 239 1393 8119 47321 275807 1607521 9369319

First 10 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)

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)
    mpz_add(p2,p2,p0)
    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]

Python

# pell_numbers.py by Xing216
def is_prime(n):
    if n == 1:
        return False
    i = 2
    while i*i <= n:
        if n % i == 0:
            return False
        i += 1
    return True
def pell(p0: int,p1: int,its: int):
    nums = [p0,p1]
    primes = {}
    idx = 2
    while len(nums) != its:
        p = 2*nums[-1]+nums[-2]
        if is_prime(p):
           primes[idx] = p
        nums.append(p)
        idx += 1
    return nums, primes
def nsw(its: int,pell_nos: list):
    nums = []
    for i in range(its):
        nums.append(pell_nos[2*i] + pell_nos[2*i+1])
    return nums
def pt(its: int, pell_nos: list): 
    nums = []
    for i in range(1,its+1):
        hypot = pell_nos[2*i+1]
        shorter_leg = sum(pell_nos[:2*i+1])
        longer_leg = shorter_leg + 1
        nums.append((shorter_leg,longer_leg,hypot))
    return nums
pell_nos, pell_primes = pell(0,1,50)
pell_lucas_nos, pl_primes = pell(2,2,10)
nsw_nos = nsw(10, pell_nos)
pythag_triples = pt(10, pell_nos)
sqrt2_approx = {}
for idx, pell_no in enumerate(pell_nos):
    numer = pell_nos[idx-1] + pell_no
    if pell_no != 0:
        sqrt2_approx[f"{numer}/{pell_no}"] = numer/pell_no
print(f"The first 10 Pell Numbers:\n {' '.join([str(_) for _ in pell_nos[:10]])}")
print(f"The first 10 Pell-Lucas Numbers:\n {' '.join([str(_) for _ in pell_lucas_nos])}")
print(f"The first 10 rational and decimal approximations of sqrt(2) ({(2**0.5):.10f}):")
print("  rational | decimal")
for rational in list(sqrt2_approx.keys())[:10]:
    print(f"{rational:>10}{sqrt2_approx[rational]:.10f}")
print("The first 7 Pell Primes:")
print(" index | Pell Prime")
for idx, prime in pell_primes.items():
    print(f"{idx:>6} | {prime}")
print(f"The first 10 Newman-Shank-Williams numbers:\n {' '.join([str(_) for _ in nsw_nos])}")
print(f"The first 10 near isosceles right triangles:")
for i in pythag_triples:
    print(f" {i}")
Output:
The first 10 Pell Numbers:
 0 1 2 5 12 29 70 169 408 985
The first 10 Pell-Lucas Numbers:
 2 2 6 14 34 82 198 478 1154 2786
The first 10 rational and decimal approximations of sqrt(2) (1.4142135624):
  rational | decimal
       1/1 ≈ 1.0000000000
       3/2 ≈ 1.5000000000
       7/5 ≈ 1.4000000000
     17/12 ≈ 1.4166666667
     41/29 ≈ 1.4137931034
     99/70 ≈ 1.4142857143
   239/169 ≈ 1.4142011834
   577/408 ≈ 1.4142156863
  1393/985 ≈ 1.4142131980
 3363/2378 ≈ 1.4142136249
The first 7 Pell Primes:
 index | Pell Prime
     2 | 2
     3 | 5
     5 | 29
    11 | 5741
    13 | 33461
    29 | 44560482149
    41 | 1746860020068409
The first 10 Newman-Shank-Williams numbers:
 1 7 41 239 1393 8119 47321 275807 1607521 9369319
The first 10 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)

Quackery

isprime is defined at Primality by trial division#Quackery.

As this method of testing for primality is not well suited to the task the Pell Primes part is limited to finding the first seven and their indices. More than that would be impractical.

  [ $ "bigrat.qky" loadfile ] now!

  [ ' [ 0 ] swap
    witheach
      [ over -1 peek 
        + join ] 
    behead drop ]             is cumsum     (   [ --> [   )

  [ dup -1 peek 2 * 
    over -2 peek + join ]     is nextterm   (   [ --> [   )

  [ over 2 - times 
      nextterm 
    swap split drop ]         is sequence   ( n [ --> [   )

  [ ' [ 0 1 ] sequence ]      is pells      (   n --> [   )

  [ ' [ 2 2 ] sequence ]      is companions (   n --> [   )

  [ [] swap 1+ 
    dup companions
    behead drop
    swap pells
    behead drop
    witheach 
      [ dip 
          [ behead 2 / ]
        join nested
        rot swap join
        swap ]
    drop ]                    is rootytwos  (   n --> [   )

  [ stack ]                   is index      (     --> s   )

  [ temp put
    1 index put
    []  ' [ 0 1 ] 
    [ 1 index tally
      over size
      temp share < while 
      nextterm
      behead drop
      index share 
      isprime until
      dup -1 peek
      dup isprime iff
        [ swap dip 
            [ index share 
              swap join 
              nested join ] ]
      else drop
      again ]
     drop 
     index release
     temp release ]           is pellprimes (   n --> [   )

  [ [] over 2 * pells
    rot times
      [ behead dip behead +
        join ] 
    nip ]                     is nsws       (   n --> [   )

  [ [] swap 1+ dup
    2 * 1+ pells
    dup cumsum
    swap rot times
      [ over i^ 2 * peek
        dup 1+ join
        over i^ 2 * 1+ peek
        join 
        dip rot nested join
        unrot ] 
    2drop behead drop ]       is nirts      (   n --> [   )

  say "Pell numbers " 10 pells echo
  cr cr
  say "Pell-Lucas's " 10 companions echo
  cr cr
  say "Approximations of sqrt(2)"
  cr
  10 rootytwos
    witheach
      [ do 2dup
        vulgar$ echo$ sp 
        10 point$ echo$ cr ]
  cr
  say "Pell Primes   "
  7 pellprimes dup
  [] swap
  witheach [ 1 peek join ] echo
  cr
  say "their indices "
  [] swap
  witheach [ 0 peek join ] echo
  cr cr
  say "NSW numbers " 10 nsws echo
  cr cr
  say "Near isosceles right triangles"
  cr
  10 nirts witheach [ echo cr ]
Output:
Pell numbers [ 0 1 2 5 12 29 70 169 408 985 ]

Pell-Lucas's [ 2 2 6 14 34 82 198 478 1154 2786 ]

Approximations of sqrt(2)
1/1 1
3/2 1.5
7/5 1.4
17/12 1.4166666667
41/29 1.4137931034
99/70 1.4142857143
239/169 1.4142011834
577/408 1.4142156863
1393/985 1.414213198
3363/2378 1.4142136249

Pell Primes   [ 2 5 29 5741 33461 44560482149 1746860020068409 ]
their indices [ 2 3 5 11 13 29 41 ]

NSW numbers [ 1 7 41 239 1393 8119 47321 275807 1607521 9369319 ]

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 ]

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)

RPL

Let's start first with a universal Pell sequence generator, which takes a smaller sequence as input:

Works with: Halcyon Calc version 4.2.8
RPL code Comment
 ≪ → n 
   ≪ IF DUP SIZE n < THEN
         LIST→ n SWAP - 
         1 SWAP START DUP2 DUP + + NEXT
         n →LIST END  
≫ ≫ 'PELLS' STO
PELLS ( { P(0)..P(m) } n -- { P(0)..P(n-1) } )
if m < n
  put P(0)..P(m) in the stack
  loop n-m times: P(j+1) = 2*P(j) + P(j-1)
  pop stack to a list
return new list

We can then generate Pell and Pell-Lucas numbers with the same function:

{ 0 1 } 25 PELLS
{ 2 2 } 25 PELLS
Output:
2: { 0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109 15994428 38613965 93222358 225058681 543339720 }
1: { 2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638 45239074 109216786 263672646 636562078 1536796802 }

Generating a sequence of rational approximations needs some code for appropriate formatting:

≪ { 0 1 } 16 PELLS → p 
   ≪ { } 2 15 FOR j 
         p j GET "'" OVER p j 1 - GET + →STR + "/" + SWAP →STR + STR→ DUP →NUM ROT ROT + SWAP + NEXT 
≫ ≫ 'TASK3' STO
Output:
{ '1/1' 1 '3/2' 1.5 '7/5' 1.4 '17/12' 1.41666666667 '41/29' 1.41379310345 '99/70' 1.41428571429 '239/169' 1.41420118343 '577/408' 1.41421568627 '1393/985' 1.41421319797 '3363/2378' 1.41421362489 '8119/5741' 1.41421355165 '19601/13860' 1.41421356421 '47321/33461' 1.41421356206 '114243/80782' 1.41421356243 }

To look for prime Pell numbers, we need a fast Pell number generator and also a prime number checker, available here:

DO 
      1 + ROT ROT DUP DUP + ROT + ROT
   UNTIL 3 PICK BPRIM? END
≫ ≫ 'PRPL' STO

≪ n { } → n primpell 
  ≪ # 0d # 1d 1 1 n START 
        PRPL 3 PICK B→R OVER 1 - R→C primpell SWAP + 'primpell' STO NEXT 
     3 DROPN primpell 
≫ ≫ 'TASK4' STO

TASK4 returns a list of complex numbers, the real part being the prime number and the imaginary part being the indice.

Output:
1: { (2,2) (5,3) (29,5) (5741,11) (33461,13) (44560482149,29) }

Unfortunately, the emulator's watchdog timer has refused to let us look beyond P29.

Generating the NSW sequence is again light work:

≪ → n
   ≪ { 0 1 } 2 n * 1 + PELLS { }
      0 n 1 - FOR j
        OVER j DUP + 1 + GET LAST 1 + GET + + NEXT
     SWAP DROP
≫ ≫ 'NSW' STO
15 NSW
Output:
1: { 1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 }

For dessert, the Pythagorean triplets can be found with:

≪ → n
   ≪ { 0 1 } 2 n * 2 + PELLS { }
      1 n FOR j
        OVER 2 j * 2 + GET LASTARG 1 - 1 SWAP SUB ∑LIST DUP 1 + 
        ROT 3 →LIST 1 →LIST + NEXT
     SWAP DROP
≫ ≫ 'PYTH3' STO
15 PYTH3

This program runs on HP-48G or newer models only, but it can be adapted for older ones by creating a homemade version of ∑LIST, which sums the items of a list, and replacing LASTARG with LAST.

Output:
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 6625109 } { 27304196 27304197 38613965 }  { 159140519 159140520 225058681 } { 927538920 927538921 1311738121 } { 5406093003 5406093004 7645370045 } { 31509019100 31509019101 44560482149 } { 183648021599 183648021600 259717522849 } }

Ruby

require 'openssl'
def prime?(n) = OpenSSL::BN.new(n).prime?

pell = Enumerator.new do |y|
  pe = [0,1]
  loop{y << pe[0]; pe << pe[1]*2 + pe.shift} 
end

pell_lucas = Enumerator.new do |y|
  pe = [2,2]
  loop{y << pe[0]; pe << pe[1]*2 + pe.shift} 
end

n = 20
puts "First #{n} Pell numbers: #{pell.first(n).to_a.inspect}"
puts "\nFirst #{n} Pell-Lucas numbers: #{pell_lucas.first(n).to_a.inspect}"

n = 15
sqrt2 = pell.each_cons(2).lazy.map{|p1,p2|  Rational(p1+p2, p2)}.take(n).to_a
puts "\nThe first #{n} rational approximations of √2 (#{Math.sqrt(2)}) are:"
sqrt2.each{|n| puts "%-16s ≈ %-18s\n"% [n, n.to_f]}

puts "\nThe first #{n} Pell primes with index are:"
primes = pell.with_index.lazy.select{|n, i|prime?(i) && prime?(n)}.first(n)
primes.each {|prime, i| puts "#{i.to_s.ljust(3)} #{prime}"}

puts "\nThe first #{n} NSW numbers:"
puts  pell.first(2*n).each_slice(2).map(&:sum).join(", ")

puts "\nFirst #{n} near isosceles right triangles:"
sum = 1
pell.take(n*2+2).each_slice(2) do |p1,p2|
  next if p1 == 0
  sum += p1   
  puts "#{sum}, #{sum+1}, #{p2}"
  sum += p2
end
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]

The first 15 rational approximations of √2 (1.4142135623730951) are:
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

The first 15 Pell primes with index are:
2   2
3   5
5   29
11  5741
13  33461
29  44560482149
41  1746860020068409
53  68480406462161287469
59  13558774610046711780701
89  4125636888562548868221559797461449
97  4760981394323203445293052612223893281
101 161733217200188571081311986634082331709
167 2964793555272799671946653940160950323792169332712780937764687561
181 677413820257085084326543915514677342490435733542987756429585398537901
191 4556285254333448771505063529048046595645004014152457191808671945330235841

The first 15 NSW numbers:
1, 7, 41, 239, 1393, 8119, 47321, 275807, 1607521, 9369319, 54608393, 318281039, 1855077841, 10812186007, 63018038201

First 15 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

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, BigRat
import "./math" for Int
import "./fmt" for Fmt

var p = List.filled(40, 0)
p[0] = 0
p[1] = 1
for (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] = 2
q[1] = 2
for (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.zero
var p1 = BigInt.one
var indices = List.filled(15, 0)
var count = 0
var index = 2
var p2
while (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 = 0
p1 = 1
var sum = 1
var i = 2
while (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)