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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

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).

Crossrefs

Cf. A010517, A041113.

Programs

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 *)

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

Extensions

More terms from Colin Barker, Nov 05 2013