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 A262026 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). 4
 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, 165, 5, 1, 22419, 5, 93, 105, 11, 1, 11, 419775, 51, 927, 21, 3, 6578829, 1, 140634693, 3, 105, 57, 5019135, 13, 1, 13, 153, 15, 313191, 123, 650783, 7, 1, 1153080099, 7, 45, 19162705353, 3, 33, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The corresponding x = x0(n) values are given by A262027(n). This is a proper subset of A033317 corresponding to its odd members. For the proof that d(n) = A007970(n), the products of Conway's 2-happy couples, see the W. Lang link under A007970. 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). 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. LINKS FORMULA x0(n)^2 - d(n)*a(n)^2 = +1 with x0(n) = 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. EXAMPLE 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], ... CROSSREFS Cf. A007970, A033317, A262027, A262028. Sequence in context: A281038 A263677 A099906 * A270390 A047787 A102668 Adjacent sequences:  A262023 A262024 A262025 * A262027 A262028 A262029 KEYWORD nonn AUTHOR Wolfdieter Lang, Oct 04 2015 STATUS approved

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Last modified February 17 12:07 EST 2020. Contains 331996 sequences. (Running on oeis4.)