A261250 One half of the even entries of A033317.

1, 2, 1, 3, 1, 90, 2, 4, 2, 1, 6, 21, 5, 12, 910, 1, 2, 3, 6, 3, 2, 160, 1, 15, 12, 1794, 7, 45, 4550, 33, 6, 1, 10, 1287, 2, 113076990, 4, 8, 4, 2, 468, 15, 1, 133500, 215, 3315, 20, 3, 9, 3, 15498, 561, 26500, 1, 60, 630, 110532, 2, 3188676, 5, 10, 5, 2, 1557945, 65, 7570212227550, 1, 14, 6, 56648, 48, 455, 30, 14127

Offset

1,2

Comments

2*a(n) = y0(n) is the positive fundamental solution satisfying the Pell equation x0(n)^2 + D(n)*y0(n)^2 = +1 with D(n) coinciding apparently with Conway's rectangular numbers r(n) = A007969(n). The corresponding x0 values are given in A262024.

For a proof of this coincidence see the W. Lang link under A007969. - Wolfdieter Lang, Oct 04 2015

Links

Table of n, a(n) for n=1..74.

Example

The [r(n), x0(n), y0(n)] values for n = 1..16 are:
[2, 3, 2], [5, 9, 4], [6, 5, 2], [10, 19, 6],
[12, 7, 2], [13, 649, 180], [14, 15, 4],
[17, 33, 8], [18, 17, 4], [20, 9, 2],
[21, 55, 12], [22, 197, 42], [26, 51, 10],
[28, 127, 24], [29, 9801, 1820], [30, 11, 2], ...

Mathematica

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2 n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
Select[DeleteCases[PellSolve /@ Range[200], {}][[All, 2]], EvenQ]/2 (* Jean-François Alcover, Aug 12 2023, using the PellSolve code given in A033317 *)

Crossrefs

Cf. A033317, A033313, A000037, A007969, A262024, A262025.

Sequence in context: A096642 A263006 A323730 * A305354 A227860 A020779

Adjacent sequences: A261247 A261248 A261249 * A261251 A261252 A261253

Keyword

nonn

Author

Wolfdieter Lang, Sep 16 2015

Status

approved