This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A254930 #11 Oct 28 2019 07:44:54
%S A254930 5,7,11,9,13,17,13,19,23,17,15,21,25,17,23,27,35,23,29,21,41,25,31,23,
%T A254930 35,29,39,43,37,31,27,49,53,33,31,37,47,41,55,59,31,45,39,49,37,35,61,
%U A254930 37,35
%N A254930 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1 (primes congruent to 1 or 7 mod 8).
%C A254930 The corresponding terms y = y2(n) are given in A254931(n).
%C A254930 There is only one fundamental solution for prime 2 (no second class exists), and this solution (x, y) has been included in (A002334(1), A002335(1)) = (2, 1).
%C A254930 The second class x sequence for the primes 1 (mod 8), which are given in A007519, is A254762, and for the primes 7 (mod 8), given in A007522, it is A254766.
%C A254930 The second class solutions give the second smallest positive integer solutions of this Pell equation.
%C A254930 For comments and the Nagell reference see A254760.
%F A254930 a(n)^2 - 2*A254931(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer solving this (generalized) Pell equation.
%F A254930 a(n) = 3*A002334(n+1) - 4*A002335(n+1), n >= 1.
%e A254930 n = 3: 11^2 - 2*7^2 = 23 = A001132(3) = A007522(2).
%e A254930 The first pairs of these second class solutions [x2(n), y2(n)] are (a star indicates primes congruent to 1 (mod 8)):
%e A254930 n A001132(n) a(n) A254931(n)
%e A254930 1 7 5 3
%e A254930 2 17 * 7 4
%e A254930 3 23 11 7
%e A254930 4 31 9 5
%e A254930 5 41 * 13 8
%e A254930 6 47 17 11
%e A254930 7 71 13 7
%e A254930 8 73 * 19 12
%e A254930 9 89 * 17 10
%e A254930 10 97 * 15 8
%e A254930 11 103 21 13
%e A254930 12 113 * 25 16
%e A254930 13 127 17 9
%e A254930 14 137 * 23 14
%e A254930 15 151 27 17
%e A254930 16 167 35 23
%e A254930 17 191 23 13
%e A254930 18 193 * 29 18
%e A254930 19 199 21 11
%e A254930 20 223 41 27
%e A254930 ...
%t A254930 Reap[For[p = 2, p < 1000, p = NextPrime[p], If[MatchQ[Mod[p, 8], 1|7], rp = Reduce[x > 0 && y > 0 && x^2 - 2 y^2 == p, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; x2 = xy[[-1, 1]] // Simplify; Print[x2]; Sow[x2]]]]][[2, 1]] (* _Jean-François Alcover_, Oct 28 2019 *)
%Y A254930 Cf. A001132, A254931, A002334, A002335, A007519, A254762, A007522, A254766, A254760.
%K A254930 nonn,easy
%O A254930 1,1
%A A254930 _Wolfdieter Lang_, Feb 12 2015