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A258684 a(n) = A041105(4n+1).

%I A258684 #40 Jul 25 2025 12:28:20
%S A258684 1,63,3905,242047,15003009,929944511,57641556673,3572846569215,
%T A258684 221458845734657,13726875588979519,850844827670995521,
%U A258684 52738652440012742783,3268945606453119057025,202621888947653368792767,12559288169148055746094529,778473244598231802889068031
%N A258684 a(n) = A041105(4n+1).
%C A258684 a(n) is a solution to the Pell equation (4*b(n))^2 - 15*a(n)^2 = 1. The corresponding b(n) are A302329(n). - _Klaus Purath_, Jul 19 2025
%H A258684 Colin Barker, <a href="/A258684/b258684.txt">Table of n, a(n) for n = 0..557</a>
%H A258684 E. Kilic, Y. T. Ulutas, and N. Omur, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Omur/omur6.html">A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters</a>, J. Int. Seq. 14 (2011) #11.5.6, table 4, k=1, t=4.
%H A258684 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (62,-1).
%F A258684 a(n) = (1/2-2/sqrt(15))*(31-8*sqrt(15))^n+((15+4*sqrt(15))*(31+8*sqrt(15))^n)/30.
%F A258684 From _Colin Barker_, Jun 07 2015: (Start)
%F A258684 a(n) = 62*a(n-1)-a(n-2).
%F A258684 G.f.: (x+1) / (x^2-62*x+1). (End)
%F A258684 E.g.f.: exp(31*x)*(15*cosh(8*sqrt(15)*x) + 4*sqrt(15)*sinh(8*sqrt(15)*x))/15. - _Stefano Spezia_, Jul 25 2025
%t A258684 a[c_, p_, n_] := Module[{},
%t A258684    l := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
%t A258684    d := Denominator[Convergents[Sqrt[c], n l]] ;
%t A258684    t := Table[d[[i + 1]], {i, p, Length[d] - 1, l}]  ;
%t A258684    Return[t];
%t A258684    ];
%t A258684 a[60, 1, 20]
%t A258684 CoefficientList[Series[(1 + x)/(x^2 - 62 x + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 08 2015 *)
%t A258684 LinearRecurrence[{62,-1},{1,63},30] (* _Harvey P. Dale_, Dec 24 2015 *)
%o A258684 (PARI) Vec((x+1)/(x^2-62*x+1) + O(x^100)) \\ _Colin Barker_, Jun 07 2015
%o A258684 (Magma) I:=[1,63]; [n le 2 select I[n] else 62*Self(n-1)-Self(n-2): n in [1..45]]; // _Vincenzo Librandi_, Jun 08 2015
%Y A258684 Cf. A041105 (denominators of continued fraction convergents to sqrt(60)).
%K A258684 nonn,easy,frac
%O A258684 0,2
%A A258684 _Gerry Martens_, Jun 07 2015