%I
%S 5,9,7,13,11,9,21,13,11,19,25,17,15,29,21,19,15,31,23,37,17,35,27,41,
%T 25,33,23,21,29,37,49,23,21,41,47,39,29,37,25,23,57,35,43,33,49,55,27,
%U 59,65,33,51,43,31,29,41,49,69,55,53,29,43,59,51,41,37,35
%N 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 {1,7} mod 8).
%C For the corresponding term y2(n) see A255248(n).
%C For the positive fundamental proper (sometimes called primitive) solutions x1(n) and y1(n) of the first class of this (generalized) Pell equation see A255235(n) and A255246(n).
%C The present solutions of this second class are the next to smallest positive ones. Note that for prime 2 only the first class exists.
%C For the derivation based on the book of Nagell see the comments on A254934 and A254938 for the primes 1 (mod 8) and 7 (mod 8) separately, where also the Nagell reference is given.
%F a(n)^2  2*A255248(n)^2 = A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.
%F a(n) = (3*A255235(n+1)  4*A255246(n+1)), n >= 1.
%e The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are
%e the prime A001132(n) is listed as first entry):
%e [7, [5, 4]], [17, [9, 7]], [23, [7, 6]],
%e [31, [13, 10]], [41, [11, 9]], [47, [9, 8]],
%e [71, [21, 16]], [73, [13, 11]], [79, [11, 10],
%e [89, [19, 15]], [97, [25, 19]], [103, [17, 14]],
%e [113, [15, 13]], [127, [29, 22]],
%e [137, [21, 17]], [151, [19, 16]],
%e [167, [15, 14]], [191, [31, 24]],
%e [193, [23, 19]], [199, [37, 28]],
%e [223, [17, 16]], [233, [35, 27]],
%e [239, [27, 22]], [241, [41, 31]], ...
%e n = 1: 5^2  2*4^2 = 25  32 = 7 = A001132(1).
%e a(3) = (3*3  4*4) = 16  9 = 7.
%Y Cf. A001132, A255248, A255235, A255246, A254936, A255233, A254930.
%K nonn
%O 1,1
%A _Wolfdieter Lang_, Feb 19 2015
%E More terms from _Colin Barker_, Feb 26 2015
