A041112 Numerators of continued fraction convergents to sqrt(65).

8, 129, 2072, 33281, 534568, 8586369, 137916472, 2215249921, 35581915208, 571525893249, 9179996207192, 147451465208321, 2368403439540328, 38041906497853569, 611038907405197432, 9814664424981012481, 157645669707101397128, 2532145379738603366529, 40671971745524755261592

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (16,1).

Formula

From Philippe Deléham, Nov 21 2008: (Start)

a(n) = 16*a(n-1) + a(n-2), with n > 1, a(0) = 8, a(1) = 129.

G.f.: (8 + x)/(1 - 16*x - x^2). (End)

E.g.f.: exp(8*x)*(8*cosh(sqrt(65)*x) + sqrt(65)*sinh(sqrt(65)*x)). - Stefano Spezia, Oct 28 2022

Mathematica

CoefficientList[Series[(8 + x)/(1 - 16 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 29 2013 *)
Numerator[Convergents[Sqrt[65], 20]] (* or *) LinearRecurrence[{16, 1}, {8, 129}, 20] (* Harvey P. Dale, Nov 12 2013 *)

Crossrefs

Cf. A010517, A041113.

Sequence in context: A027951 A041115 A348546 * A348207 A356914 A364986

Adjacent sequences: A041109 A041110 A041111 * A041113 A041114 A041115

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

More terms from Colin Barker, Nov 05 2013

Status

approved