A378849 -id:A378849 - cp's OEIS frontend

A378850 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 = -2.

1, 1, 1, 4, 13, 31, 73, 190, 505, 1316, 3431, 9065, 24122, 64325, 172082, 462344, 1246685, 3371135, 9140289, 24847422, 67708743, 184906614, 505986933, 1387240401, 3810083424, 10481797131, 28880894706, 79692785251, 220203155689, 609242057143, 1687666776031

Offset: 0

Emely Hanna Li Lobnig, Dec 09 2024

nonn

			For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0).

Cf. A378849, A116411, A071879, A026325.

a := n -> hypergeom([4/3, 5/3, (1-n)/3, (2-n)/3, -n/3], [1/3, 2/3, 5/2, 2], -27/4):
seq(simplify(a(n)), n = 0..30);  # Peter Luschny, Dec 18 2024

a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+3, k+1)/(2*k+3)) \\ Thomas Scheuerle, Dec 09 2024

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.

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 75, 204, 561, 1499, 4001, 10814, 29364, 79704, 216672, 590764, 1614421, 4419049, 12116139, 33277722, 91546143, 252209535, 695803659, 1922166420, 5316714156, 14723570406, 40820144106, 113293243636, 314759548879, 875342190283, 2436582442381

Offset: 0

Emely Hanna Li Lobnig, Dec 23 2024

nonn

			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.

Cf. A071879, A116411, A378849, A378850.

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

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

a(n) ~ 23 * (1 + 3/2^(2/3))^(n + 3/2) / (4 * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2025

More terms from Jinyuan Wang, Jan 07 2025

cp's OEIS Frontend

A378850 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 = -2.

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