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%I #19 Oct 24 2015 12:47:14
%S 1,3,1,3,1,39,5,1,5,273,3,1,3,531,7,1,7,69,1,5967,413,3,9,1,9,3,21,
%T 165,5,1,22419,5,93,105,11,1,11,419775,51,927,21,3,6578829,1,
%U 140634693,3,105,57,5019135,13,1,13,153,15,313191,123,650783,7,1,1153080099,7,45,19162705353,3,33,5
%N The positive odd fundamental solutions y = y0(n) for the Pell equation x^2 - d*y^2 = +1. It turns out that d = d(n) coincides with A007970(n).
%C The corresponding x = x0(n) values are given by A262027(n).
%C This is a proper subset of A033317 corresponding to its odd members.
%C For the proof that d(n) = A007970(n), the products of Conway's 2-happy couples, see the W. Lang link under A007970.
%C For the positive even fundamental solutions y = y0(n) of x^2 - d*y^2 = 1, where d = d(n) coincides with A007969(n) see 2*A261250(n).
%C If d(n) = A007970(n) is odd (necessarily congruent to 3 modulus 4) then x0(n) is even, and if d(n) is even (necessarily congruent to 0 modulus 8) then x0 is odd.
%F x0(n)^2 - d(n)*a(n)^2 = +1 with x0(n) =
%F A262027(n) and d(n) = A007970(n). (x0(n), y0(n) = a(n)) are the positive fundamental solutions of this Pell equation x^2 - d*y^2 = +1 with odd y = y0.
%e The first triples [d(n), x0(n), y0(n)] are: [3,2,1], [7,8,3], [8,3,1], [11,10,3], [15,4,1], [19,170,39], [23,24,5], [24,5,1], [27,26,5], [31,1520,273], [32,17,3], [35,6,1], [40,19,3], [43,3482,531], [47,48,7], [48,7,1], [51,50,7], [59,530,69], [63,8,1], [67,48842,5967], [71,3480,413], [75,26,3], [79,80,9], [80,9,1], [83,82,9], [87,28,3], [88,197,21], [91,1574,165], [96,49,5], [99,10,1], [103,227528,22419], ...
%Y Cf. A007970, A033317, A262027, A262028.
%K nonn
%O 1,2
%A _Wolfdieter Lang_, Oct 04 2015
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