A362067 Sum of successive Fibonacci numbers F(n) : a(n) = Sum_{k = 0..n} F(n+k).

0, 2, 6, 18, 50, 136, 364, 966, 2550, 6710, 17622, 46224, 121160, 317434, 831430, 2177322, 5701290, 14927768, 39083988, 102327390, 267903350, 701391022, 1836283246, 4807480608, 12586194000, 32951158706, 86267374854, 225851115906, 591286215650, 1548007923880

Offset: 0

Author

Philippe Deléham, Apr 07 2023

Keywords

• nonn
• easy

Examples

			a(n) are the row sums of the triangle T(n,k) (A199334):
  0;
  1, 1;
  1, 2,  3;
  2, 3,  5,  8;
  3, 5,  8, 13, 21;
  5, 8, 13, 21, 34, 55;
  ...
T(n,k) = T(n,k-1) + T(n-1, k-1); T(n,0) = A000045(n).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Crossrefs

Cf. A000032, A000045, A094292, A199334.

Programs

Mathematica

A362067[n_] := Fibonacci[n+1]*(LucasL[n+1] - 1); Array[A362067, 50, 0] (* or *)
LinearRecurrence[{4, -3, -2, 1}, {0, 2, 6, 18}, 50] (* Paolo Xausa, Jun 10 2024 *)

Formula

a(n) = 4*a(n-1)-3*a(n-2)-2*a(n-3)+a(n-4), a(0)=0, a(1)=2, a(2)=6, a(3)=18.

G.f.: 2*x*(1-x)/((1-3*x+x^2)*(1-x-x^2)).

a(n) = A000045(2n+2) - A000045(n+1).

a(n) = 2 * A094292(n+1). - Alois P. Heinz, Apr 07 2023

a(n) = A000045(n+1)*(A000032(n+1) - 1). - Paolo Xausa, Jun 10 2024