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 A116391 #15 Sep 08 2022 08:45:24
%S A116391 1,0,3,2,11,14,47,78,217,408,1039,2086,5065,10560,24931,53194,123403,
%T A116391 267222,612903,1340222,3050679,6714946,15205967,33622158,75864835,
%U A116391 168275790,378743151,841959974,1891648931,4211866694,9450828951
%N A116391 Expansion of 1/((1+x)*(sqrt(1-4*x^2)-x)).
%H A116391 G. C. Greubel, <a href="/A116391/b116391.txt">Table of n, a(n) for n = 0..1000</a>
%F A116391 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..k} Sum_{i=0..floor((n-k)/2)} (-1)^(k-j)*C(k,j)*C(i+(j-1)/2,i)*C(j,n-k-2i)*4^i.
%F A116391 Conjecture D-finite with recurrence: n*a(n) +(n)*a(n-1) +3*(-3*n+4)*a(n-2) +3*(-3*n+4)*a(n-3) +20*(n-3)*a(n-4) +20*(n-3)*a(n-5)=0. - _R. J. Mathar_, Jan 23 2020
%F A116391 a(n) ~ 5^(n/2)/(1+sqrt(5)). - _Vaclav Kotesovec_, Nov 19 2021
%t A116391 CoefficientList[Series[1/((1+x)(Sqrt[1-4(x^2) ]-x)),{x,0,40}],x] (* _Harvey P. Dale_, Sep 25 2018 *)
%o A116391 (PARI) my(x='x+O('x^30)); Vec(1/((1+x)*(sqrt(1-4*x^2)-x))) \\ _G. C. Greubel_, May 23 2019
%o A116391 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/((1+x)*(Sqrt(1-4*x^2)-x)) )); // _G. C. Greubel_, May 23 2019
%o A116391 (Sage) (1/((1+x)*(sqrt(1-4*x^2)-x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 23 2019
%Y A116391 Diagonal sums of A116389.
%K A116391 easy,nonn
%O A116391 0,3
%A A116391 _Paul Barry_, Feb 12 2006