# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a255233 Showing 1-1 of 1 %I A255233 #12 Feb 25 2015 11:32:58 %S A255233 5,7,13,9,21,11,17,29,19,15,31,37,17,27,33,23,29,21,41,47,37,23,43,33, %T A255233 49,55,51,31,41,69,53,29,43,59,35,31,45,61,41,67,85,57,47,63,43,53,35, %U A255233 75,93,37,71,61,83,47,89,39,73,53,63,79,49,85,69,97,103,109,55 %N A255233 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8). %C A255233 The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007522(n) = -(1 + A139487(n)*8) is given in 2*A255234(n). %C A255233 For comments and the Nagell reference see A254938. %F A255233 a(n)^2 - 2*(2*A255234(n))^2 = -A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation. %F A255233 a(n) = -(3*A254938(n) - 4*2*A255232(n)), n >= 1. %e A255233 The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007522(n) appears as first entry): %e A255233 [7, [5, 4]], [23, [7, 6]], [31, [13, 10]], %e A255233 [47, [9, 8]], [71, [21, 16]], [79, [11, 10]], [103, [17, 14]], [127, [29, 22]], %e A255233 [151, [19, 16]], [167, [15, 14]], %e A255233 [191, [31, 24]], [199, [37, 28]], %e A255233 [223, [17, 16]], [239, [27, 22]], %e A255233 [263, [33, 26]], [271, [23, 20]], %e A255233 [311, [29, 24]], [359, [21, 20]], %e A255233 [367, [41, 32]], [383, [47, 36]], %e A255233 [431, [37, 30]], [439, [23, 22]], %e A255233 [463, [43, 34]], [479, [33, 28]], ... %e A255233 n= 4: 9^2 - 2*(2*4)^2 = -47 = -A007522(4). %e A255233 a(4) = -(3*5 - 4*(2*3)) = 24 - 15 = 9. %Y A255233 Cf. A007522, A139487, 2*A255234, A254938, 2*A255232, A255247, A254936. %K A255233 nonn,look,easy %O A255233 1,1 %A A255233 _Wolfdieter Lang_, Feb 18 2015 %E A255233 More terms from _Colin Barker_, Feb 23 2015 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE