A140833 - OEIS

%I #21 Aug 07 2022 14:56:01

%S 0,2,2,6,6,16,16,42,42,110,110,288,288,754,754,1974,1974,5168,5168,

%T 13530,13530,35422,35422,92736,92736,242786,242786,635622,635622,

%U 1664080,1664080,4356618,4356618,11405774,11405774,29860704,29860704,78176338,78176338

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

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

%H Matthew House, <a href="/A140833/b140833.txt">Table of n, a(n) for n = 0..4760</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).

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

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

%F 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

%F a(n) = -a(-1-n) for all n in Z. - _Michael Somos_, Nov 01 2016

%F a(n) = 2*A000045(ceiling(n/2)*2). - _Alois P. Heinz_, Nov 02 2016

%e a(3) = 2+(-1)+1+0+1+1+2=6.

%e G.f. = 2*x + 2*x^2 + 6*x^3 + 6*x^4 + 16*x^5 + 16*x^6 + 42*x^7 + ...

%p a:= n-> 2*(<<0|1>, <1|1>>^(ceil(n/2)*2))[1,2]:

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Nov 02 2016

%t a[ n_] := 2 Fibonacci[ n + Mod[n, 2]]; (* _Michael Somos_, Nov 01 2016 *)

%t LinearRecurrence[{0,3,0,-1},{0,2,2,6},50] (* _Harvey P. Dale_, Aug 07 2022 *)

%o (PARI) {a(n) = 2 * fibonacci(n + n%2)}; /* _Michael Somos_, Nov 01 2016 */

%Y Cf. A000045, A025169, A001906.

%K nonn,easy

%O 0,2

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

%E a(21)-a(22) corrected by _Matthew House_, Nov 01 2016