A140833 Sum of Fibonacci numbers between F(-n)....F(n), inclusive.

0, 2, 2, 6, 6, 16, 16, 42, 42, 110, 110, 288, 288, 754, 754, 1974, 1974, 5168, 5168, 13530, 13530, 35422, 35422, 92736, 92736, 242786, 242786, 635622, 635622, 1664080, 1664080, 4356618, 4356618, 11405774, 11405774, 29860704, 29860704, 78176338, 78176338

Offset: 0

Author

Carey W. Strutz (cwstrutz(AT)excite.com), Jul 18 2008

Keywords

• nonn
• easy

Comments

a(2n)/a(2n+1) converges to ((((sqrt 5)-1)/2)^2).

Examples

			a(3) = 2+(-1)+1+0+1+1+2=6.
G.f. = 2*x + 2*x^2 + 6*x^3 + 6*x^4 + 16*x^5 + 16*x^6 + 42*x^7 + ...
		

Links

• Matthew House, Table of n, a(n) for n = 0..4760
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A000045, A025169, A001906.

Programs

Maple

a:= n-> 2*(<<0|1>, <1|1>>^(ceil(n/2)*2))[1,2]:
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 02 2016

Mathematica

a[ n_] := 2 Fibonacci[ n + Mod[n, 2]]; (* Michael Somos, Nov 01 2016 *)
LinearRecurrence[{0,3,0,-1},{0,2,2,6},50] (* Harvey P. Dale, Aug 07 2022 *)

PARI

{a(n) = 2 * fibonacci(n + n%2)}; /* Michael Somos, Nov 01 2016 */

Formula

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

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

G.f.: 2x(1+x)/((1-x-x^2)(1+x-x^2)). a(n)=2*A094966(n) = A000045(n+2)-A039834(n-1). - R. J. Mathar, Oct 30 2008

a(n) = -a(-1-n) for all n in Z. - Michael Somos, Nov 01 2016

a(n) = 2*A000045(ceiling(n/2)*2). - Alois P. Heinz, Nov 02 2016

Extensions

a(21)-a(22) corrected by Matthew House, Nov 01 2016