A368375 Arises from enumeration of a certain class of partial knight's paths on the square grid.

0, 1, 0, 2, 1, 9, 10, 42, 64, 213, 400, 1180, 2510, 6918, 15842, 42134, 100966, 263894, 650688, 1687746, 4238855, 10968196, 27888906, 72187356, 185141385, 480025465, 1238940890, 3219591600, 8350054339, 21752631006, 56634075418, 147899071110, 386293111740

Offset: 0

Author

N. J. A. Sloane, Feb 17 2024

Keywords

• nonn

Comments

It would be nice to have a more precise definition.

Links

• 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 2.1.

Crossrefs

Cf. A005220, A005221.

Programs

Mathematica

aa = (1 + 2z + Sqrt[1 - 4z + 4z^2 - 4z^4] - Sqrt[2]*Sqrt[1 - 4z^2 - 2z^4 + (2z + 1)*Sqrt[1 - 4z + 4z^2 - 4z^4]])/(4z^2);
a1 = z^2 aa^2 / (1 - z aa);
gf = (z (u z + 1) aa + u z a1 - u^2) / (u^4 z + u^3 z^2 + u z^2 - u^2 + z);
Table[SeriesCoefficient[gf,{u,0,2},{z,0,n}], {n,0,30}] (* Andrei Zabolotskii, Jul 25 2025 *)

Formula

G.f.: (1 - (1+x) * A(x) + (x+x^4) * A(x)^2) / (x * (x * A(x) - 1)), where A(x) is the g.f. of A005220. - Andrei Zabolotskii, Jul 25 2025

Extensions

Terms a(10) and beyond from Andrei Zabolotskii, Jul 25 2025