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

1, 63, 3905, 242047, 15003009, 929944511, 57641556673, 3572846569215, 221458845734657, 13726875588979519, 850844827670995521, 52738652440012742783, 3268945606453119057025, 202621888947653368792767, 12559288169148055746094529, 778473244598231802889068031

Offset: 0

Author

Gerry Martens, Jun 07 2015

Keywords

• nonn
• easy
• frac

Comments

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

Links

• Colin Barker, Table of n, a(n) for n = 0..557
• E. Kilic, Y. T. Ulutas, and N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, table 4, k=1, t=4.
• Index entries for linear recurrences with constant coefficients, signature (62,-1).

Crossrefs

Cf. A041105 (denominators of continued fraction convergents to sqrt(60)).

Programs

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

Mathematica

a[c_, p_, n_] := Module[{},
   l := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Denominator[Convergents[Sqrt[c], n l]] ;
   t := Table[d[[i + 1]], {i, p, Length[d] - 1, l}]  ;
   Return[t];
   ];
a[60, 1, 20]
CoefficientList[Series[(1 + x)/(x^2 - 62 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 08 2015 *)
LinearRecurrence[{62,-1},{1,63},30] (* Harvey P. Dale, Dec 24 2015 *)

PARI

Vec((x+1)/(x^2-62*x+1) + O(x^100)) \\ Colin Barker, Jun 07 2015

Formula

a(n) = (1/2-2/sqrt(15))*(31-8*sqrt(15))^n+((15+4*sqrt(15))*(31+8*sqrt(15))^n)/30.

From Colin Barker, Jun 07 2015: (Start)

a(n) = 62*a(n-1)-a(n-2).

G.f.: (x+1) / (x^2-62*x+1). (End)

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