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URL: https://oeis.org/A261250

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A261250
One half of the even entries of A033317.
7
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
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
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Sep 16 2015
STATUS
approved