A245561 a(n) = 5^n - ( (sqrt(5)*phi)^n + (sqrt(5)/phi)^n ) + 1, where phi = golden ratio A001622.

0, 1, 11, 76, 451, 2501, 13376, 70001, 361251, 1846876, 9381251, 47437501, 239109376, 1202500001, 6037656251, 30279296876, 151725781251, 759820312501, 3803412109376, 19032656250001, 95219707031251, 476302685546876, 2382252050781251, 11913932617187501

Offset

0,3

References

Roger L. Bagula, email message, Aug 08 2014.

Links

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

Index entries for linear recurrences with constant coefficients, signature (11,-40,55,-25).

Formula

a(n) = 5^n - A020876(n) + 1.

G.f.: -x*(5*x^2-1)/((1-5*x+5*x^2)*(x-1)*(5*x-1)). - Vincenzo Librandi, Aug 08 2014

Maple

g:=n->simplify(rationalize(simplify(expand( (sqrt(5)*p)^n + (sqrt(5)*q)^n ))); # A020876
h:=n->5^n-g(n)+1;
[seq(h(n), n=0..40)];

Mathematica

CoefficientList[Series[-x (5 x^2 - 1)/((1 - 5 x + 5 x^2) (x - 1)(5 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 08 2014 *)
LinearRecurrence[{11, -40, 55, -25}, {0, 1, 11, 76}, 30] (* Harvey P. Dale, Nov 05 2017 *)

Prog

(Magma) [5^n + 1 - Floor(((5+Sqrt(5))/2)^n+((5-Sqrt(5))/2)^n): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014

Crossrefs

Cf. A020876, A001622.

Sequence in context: A302382 A034269 A256597 * A056914 A232032 A272395

Adjacent sequences: A245558 A245559 A245560 * A245562 A245563 A245564

Keyword

nonn

Author

N. J. A. Sloane, Aug 08 2014

Status

approved