A101353 a(n) = Sum_{k=0..n} (2^k + Fibonacci(k)).

1, 4, 9, 19, 38, 75, 147, 288, 565, 1111, 2190, 4327, 8567, 16992, 33753, 67131, 133654, 266323, 531051, 1059520, 2114861, 4222959, 8434974, 16852239, 33675823, 67305280, 134535537, 268949683, 537702950, 1075088091, 2149661955, 4298491872, 8595637477

Offset: 0

Author

Jorge Coveiro, Dec 25 2004

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

Crossrefs

Cf. A117591 (first differences). - R. J. Mathar, Feb 06 2010

Programs

Maple

seq(sum(2^x+fibonacci(x),x=0..a),a=0..30);

Mathematica

Accumulate[Table[2^k+Fibonacci[k],{k,0,40}]] (* or *) LinearRecurrence[{4,-4,-1,2},{1,4,9,19},40] (* Harvey P. Dale, Aug 17 2025 *)

PARI

Vec((1-3*x^2)/((1-x)*(2*x-1)*(x^2+x-1)) + O(x^40)) \\ Colin Barker, Nov 03 2016

Formula

Fibonacci(n+2) + 2^(n+1) + 2. - Ralf Stephan, May 16 2007

a(n)= 4*a(n-1) -4*a(n-2) -a(n-3) +2*a(n-4). G.f.: (1-3*x^2)/((1-x) * (2*x-1) * (x^2+x-1)). - R. J. Mathar, Feb 06 2010

a(n) = (-2+2^(1+n)+(2^(-1-n)*((1-sqrt(5))^n*(-3+sqrt(5))+(1+sqrt(5))^n*(3+sqrt(5))))/sqrt(5)). - Colin Barker, Nov 03 2016