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%I #14 Aug 12 2023 12:35:54
%S 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,
%T 4550,33,6,1,10,1287,2,113076990,4,8,4,2,468,15,1,133500,215,3315,20,
%U 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
%N One half of the even entries of A033317.
%C 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.
%C For a proof of this coincidence see the W. Lang link under A007969. - _Wolfdieter Lang_, Oct 04 2015
%e The [r(n), x0(n), y0(n)] values for n = 1..16 are:
%e [2, 3, 2], [5, 9, 4], [6, 5, 2], [10, 19, 6],
%e [12, 7, 2], [13, 649, 180], [14, 15, 4],
%e [17, 33, 8], [18, 17, 4], [20, 9, 2],
%e [21, 55, 12], [22, 197, 42], [26, 51, 10],
%e [28, 127, 24], [29, 9801, 1820], [30, 11, 2], ...
%t 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]}];
%t Select[DeleteCases[PellSolve /@ Range[200], {}][[All, 2]], EvenQ]/2 (* _Jean-François Alcover_, Aug 12 2023, using the PellSolve code given in A033317 *)
%Y Cf. A033317, A033313, A000037, A007969, A262024, A262025.
%K nonn
%O 1,2
%A _Wolfdieter Lang_, Sep 16 2015
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