A101351 a(n) = 2^n-1 + Fibonacci(n).

2, 4, 9, 18, 36, 71, 140, 276, 545, 1078, 2136, 4239, 8424, 16760, 33377, 66522, 132668, 264727, 528468, 1055340, 2108097, 4212014, 8417264, 16823583, 33629456, 67230256, 134414145, 268753266, 537385140, 1074573863, 2148829916, 4297145604, 8593459169, 17185572070

Offset

1,1

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

Formula

From R. J. Mathar, Feb 06 2010: (Start)

a(n) = 4*a(n-1) -4*a(n-2) -a(n-3) +2*a(n-4).

G.f.: x*(2-4*x+x^2)/((x-1) * (2*x-1) * (1-x-x^2)). (End)

a(n) = ((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^n*sqrt(5)) + 2^n - 1. - Colin Barker, Nov 02 2016

E.g.f.: exp(x)*(exp(x) - 1) + 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Jun 05 2026

Maple

seq(2^x-1+fibonacci(x), x=1..30);

Mathematica

Table[2^n-1+Fibonacci[n], {n, 30}] (* Harvey P. Dale, Aug 24 2012 *)
(* Alternative: *)
LinearRecurrence[{4, -4, -1, 2}, {2, 4, 9, 18}, 30] (* Harvey P. Dale, Aug 24 2012 *)

Prog

(SageMath) [gaussian_binomial(n, 1, 2)+fibonacci (n) for n in range(1, 31)] # Zerinvary Lajos, May 29 2009
(PARI) Vec(x*(2-4*x+x^2)/((1-x)*(1-2*x)*(1-x-x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016

Crossrefs

Cf. A000045, A033138.

Sequence in context: A103321 A138196 A298404 * A293333 A111662 A253585

Adjacent sequences: A101348 A101349 A101350 * A101352 A101353 A101354

Keyword

nonn,easy

Author

Jorge Coveiro, Dec 25 2004

Extensions

Offset changed to 1 by Colin Barker, Nov 02 2016

Status

approved