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 A117186 #12 Jul 09 2021 07:39:00
%S A117186 1,2,5,9,21,38,86,157,349,642,1410,2610,5682,10572,22860,42717,91869,
%T A117186 172298,368906,694054,1480486,2793012,5938740,11230834,23813746,
%U A117186 45131348,95462996,181268292,382594884,727747608,1533053976
%N A117186 Expansion of (1+x)c(x^2)/((1-xc(x^2))*sqrt(1-4x^2)), c(x) the g.f. of A000108.
%C A117186 Row sums of triangle A117184.
%F A117186 G.f.: (1+x)(sqrt(1-4x^2)+2x-1)/(2x(1-2x)*sqrt(1-4x^2)); a(n)=sum{k=0..n, C(n+1,(n+k)/2+1)(1+(-1)^(n-k))/2+C(n,(n+k)/2+1/2)(1-(-1)^(n-k))/2}.
%F A117186 G.f.: (1+x)(1+2x-sqrt(1-4x^2))/(2x(1-4x^2)); a(n)=(3*2^n-binomial(2*floor((n+1)/2),floor((n+1)/2)))/2; - _Paul Barry_, Jan 20 2008
%F A117186 Conjecture: a(n) = A058622(n) + A058622(n+1). [_R. J. Mathar_, Nov 21 2008]
%F A117186 Conjecture: -(n+1)*a(n) +(n+1)*a(n-1) +2*(3*n-2)*a(n-2) -4*n*a(n-3) +8*(3-n)*a(n-4)=0. - _R. J. Mathar_, Nov 15 2011
%K A117186 easy,nonn
%O A117186 0,2
%A A117186 _Paul Barry_, Mar 01 2006