Problem Statement

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

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

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

Let's start with the solution:

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


   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=: 2 >. +: %&Pn ]
   QN i.11
2 2 6 14 34 82 198 478 1154 2786 6726


   }.(%~ _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


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


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


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


   }.(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