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 A257515 #19 May 01 2020 21:23:40
%S A257515 1,0,1,2,2,4,9,12,26,48,90,172,348,664,1349,2680,5438,10976,22510,
%T A257515 45900,94700,195032,404442,838824,1748308,3646368,7632628,15994232,
%U A257515 33606168,70699504,149050669,314625264,665280246,1408436672,2986069782,6337988876
%N A257515 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at odd level.
%H A257515 Andrew Howroyd, <a href="/A257515/b257515.txt">Table of n, a(n) for n = 0..1000</a>
%F A257515 G.f.: (1-2*x^3-sqrt((1-2x^3)*(1-4*x^2-2*x^3)))/(2*x^2*(1-2*x^3)).
%F A257515 Conjecture: (n+2)*a(n) +(n+1)*a(n-1) +(n+4)*a(n-2) +4*(-2*n+3)*a(n-3) +4*(-6*n+17)*a(n-4) +4*(-3*n+10)*a(n-5) +4*(3*n-11)*a(n-6) +4*(11*n-50)*a(n-7) +20*(n-6)*a(n-8)=0. - _R. J. Mathar_, Jun 07 2016
%e A257515 For n=6 we have 9 paths: UDUDUD, H3H3 (4 options), UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).
%t A257515 CoefficientList[Series[(1-2*x^3-Sqrt[(1-2x^3)*(1-4*x^2-2*x^3)])/(2*x^2*(1-2*x^3)), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Apr 28 2015 *)
%o A257515 (Maxima)
%o A257515 a(n):=sum((binomial(2*m,m)/(m+1)*(if mod(n+m,3)=0 then 2^((n-2*m)/3)* binomial((m+n)/3,m) else 0)),m,0,n); /* _Vladimir Kruchinin_, Mar 07 2016 */
%o A257515 (PARI) seq(n)={Vec((1-2*x^3-sqrt((1-2*x^3)*(1-4*x^2-2*x^3) + O(x^(3+n))))/(2*x^2*(1-2*x^3)))} \\ _Andrew Howroyd_, May 01 2020
%Y A257515 Cf. A090344, A086622, A257178, A257388, A007317, A064613, A104455, A104498, A257389.
%K A257515 nonn
%O A257515 0,4
%A A257515 _José Luis Ramírez Ramírez_, Apr 27 2015
%E A257515 Terms a(31) and beyond from _Andrew Howroyd_, May 01 2020