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).

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

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

Table of n, a(n) for n=1..66.

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 A341472 A047787

Adjacent sequences: A262023 A262024 A262025 * A262027 A262028 A262029

Keyword

nonn

Author

Wolfdieter Lang, Oct 04 2015

Status

approved