This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A376785 #14 Jun 29 2025 17:57:35
%S A376785 1,3,5,8,19,46,98,201,429,937,2024,4325,9260,19916,42841,91999,197485,
%T A376785 424160,911255,1957402,4203998,9029425,19394681,41658577,89478064,
%U A376785 192188361,412801176,886657848,1904452689,4090568027,8786123349,18871711384,40534539675,87064092870
%N A376785 Expansion of (1 + x - x^3)/((1 + x - x^3)^2 - 4*x).
%H A376785 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,2,0,-1).
%F A376785 a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) - a(n-6).
%F A376785 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k+1,2*k+1).
%t A376785 CoefficientList[Series[(1+x-x^3)/((1+x-x^3)^2-4x),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,2,2,0,-1},{1,3,5,8,19,46},40] (* _Harvey P. Dale_, Jun 29 2025 *)
%o A376785 (PARI) my(N=40, x='x+O('x^N)); Vec((1+x-x^3)/((1+x-x^3)^2-4*x))
%o A376785 (PARI) a(n) = sum(k=0, n\3, binomial(2*n-4*k+1, 2*k+1));
%Y A376785 Cf. A099511, A376786.
%Y A376785 Cf. A375278, A375279, A376717.
%K A376785 nonn
%O A376785 0,2
%A A376785 _Seiichi Manyama_, Oct 04 2024