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

%I A140833 #21 Aug 07 2022 14:56:01
%S A140833 0,2,2,6,6,16,16,42,42,110,110,288,288,754,754,1974,1974,5168,5168,
%T A140833 13530,13530,35422,35422,92736,92736,242786,242786,635622,635622,
%U A140833 1664080,1664080,4356618,4356618,11405774,11405774,29860704,29860704,78176338,78176338
%N A140833 Sum of Fibonacci numbers between F(-n)....F(n), inclusive.
%C A140833 a(2n)/a(2n+1) converges to ((((sqrt 5)-1)/2)^2).
%H A140833 Matthew House, <a href="/A140833/b140833.txt">Table of n, a(n) for n = 0..4760</a>
%H A140833 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).
%F A140833 a(2n-1) = a(2n).
%F A140833 a(n) = 3*a(n-2) - a(n-4).
%F A140833 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 A140833 a(n) = -a(-1-n) for all n in Z. - _Michael Somos_, Nov 01 2016
%F A140833 a(n) = 2*A000045(ceiling(n/2)*2). - _Alois P. Heinz_, Nov 02 2016
%e A140833 a(3) = 2+(-1)+1+0+1+1+2=6.
%e A140833 G.f. = 2*x + 2*x^2 + 6*x^3 + 6*x^4 + 16*x^5 + 16*x^6 + 42*x^7 + ...
%p A140833 a:= n-> 2*(<<0|1>, <1|1>>^(ceil(n/2)*2))[1,2]:
%p A140833 seq(a(n), n=0..40);  # _Alois P. Heinz_, Nov 02 2016
%t A140833 a[ n_] := 2 Fibonacci[ n + Mod[n, 2]]; (* _Michael Somos_, Nov 01 2016 *)
%t A140833 LinearRecurrence[{0,3,0,-1},{0,2,2,6},50] (* _Harvey P. Dale_, Aug 07 2022 *)
%o A140833 (PARI) {a(n) = 2 * fibonacci(n + n%2)}; /* _Michael Somos_, Nov 01 2016 */
%Y A140833 Cf. A000045, A025169, A001906.
%K A140833 nonn,easy
%O A140833 0,2
%A A140833 Carey W. Strutz (cwstrutz(AT)excite.com), Jul 18 2008
%E A140833 a(21)-a(22) corrected by _Matthew House_, Nov 01 2016