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A140833 Sum of Fibonacci numbers between F(-n)....F(n), inclusive. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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).

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

EXAMPLE

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 + ...

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 *)

PROG

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

CROSSREFS

Cf. A000045, A025169, A001906.

Sequence in context: A306007 A287603 A034422 * A257389 A071908 A011260

Adjacent sequences:  A140830 A140831 A140832 * A140834 A140835 A140836

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified November 25 14:47 EST 2020. Contains 338625 sequences. (Running on oeis4.)