A368377 Arises from enumeration of a certain class of zig-zag knight's paths on the square grid.

0, 0, 0, 0, 1, 0, 2, 0, 6, 1, 15, 4, 37, 14, 91, 44, 222, 129, 541, 364, 1319, 1000, 3219, 2696, 7869, 7172, 19273, 18892, 47299, 49398, 116317, 128444, 286624, 332552, 707679, 858168, 1750588, 2208898, 4338314, 5674380, 10769893, 14554398, 26780522, 37286820

Offset

0,7

Comments

It would be nice to have a more precise definition.

Links

Table of n, a(n) for n=0..43.

Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022). Also arXiv:2206.12087 [math.CO], Jan 2023. See Section 2.2.

Formula

G.f.: (x + x^2 * R(x) + R(x)^2) * R(x)^2 / x^3 = F(x) * R(x), where R(x) = x * (A(x^2) - 1), A(x) is the g.f. of A004148, and F(x) is the g.f. of A368376. - Andrei Zabolotskii, Jul 25 2025

Mathematica

r = (1 - z^4 - z^2 - Sqrt[z^8 - 2z^6 - z^4 - 2z^2 + 1]) / (2z^3);
gf = r (u^2 z + u z^2 + 1) / (z^3 (1 - r u));
Table[SeriesCoefficient[gf, {u, 0, 3}, {z, 0, n}], {n, 0, 50}] (* Andrei Zabolotskii, Jul 25 2025 *)

Crossrefs

Cf. A144366, A368376.

Sequence in context: A393079 A092158 A051831 * A383568 A119883 A020853

Adjacent sequences: A368374 A368375 A368376 * A368378 A368379 A368380

Keyword

nonn

Author

N. J. A. Sloane, Feb 18 2024

Extensions

Terms a(16) and beyond from Andrei Zabolotskii, Jul 25 2025

Status

approved