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 A379462 #14 Jan 28 2025 03:48:25
%S A379462 1,1,1,4,13,31,75,204,561,1499,4001,10814,29364,79704,216672,590764,
%T A379462 1614421,4419049,12116139,33277722,91546143,252209535,695803659,
%U A379462 1922166420,5316714156,14723570406,40820144106,113293243636,314759548879,875342190283,2436582442381
%N A379462 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -2), and staying on or above y = -3.
%F A379462 a(n) = Sum_{k=0..floor(n/3)} 2*binomial(n, k*3)*(binomial(3*k+3, k)/(k+2) - binomial(3*k, k-1)/(k+1)). - _Thomas Scheuerle_, Jan 07 2025
%F A379462 a(n) ~ 23 * (1 + 3/2^(2/3))^(n + 3/2) / (4 * sqrt(3*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jan 15 2025
%e A379462 For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0). An example of a path with these steps, but not staying on or above y = -3, is for n=6: DDUUUU.
%o A379462 (PARI) lista(nn) = my(v=vector(nn+5), w); print1(v[4]=1); for(n=1, nn, w=v; for(i=1, n+3, w[i]+=v[i+2]; w[i+1]+=v[i]); v=w; print1(", ", v[4])); \\ _Jinyuan Wang_, Jan 07 2025
%Y A379462 Cf. A071879, A116411, A378849, A378850.
%K A379462 nonn
%O A379462 0,4
%A A379462 _Emely Hanna Li Lobnig_, Dec 23 2024
%E A379462 More terms from _Jinyuan Wang_, Jan 07 2025