A337906 The number of walks of n steps on the hexagonal lattice that start at the origin and end at the non-adjacent vertex (2,0).

1, 6, 34, 200, 1095, 6230, 34636, 195552, 1099140, 6218520, 35210736, 200099328, 1139401263, 6504768270, 37211008120, 213311576192, 1225053737622, 7047867363108, 40612374024300, 234376628718960, 1354498970905080, 7838134441085520, 45412879702456800, 263417461793120000

Offset: 2

Author

R. J. Mathar, Sep 29 2020

Keywords

• nonn
• walk

Examples

			There is a(2)=1 path with 2 steps: RR, where R=(1,0), L=(-1,0), U=(1/2,sqrt(3)/2), u=(-1/2,sqrt(3)/2), D=(1/2,-sqrt(3)/2), d=(-1/2,-sqrt(3)/2).
There are a(3)=6 paths with 3 steps: RUD, RDU, DRU, DUR, URD, UDR.
		

Crossrefs

Cf. A002898 (returns to origin), A337905, A337907.

Programs

Maple

# see A337905

Mathematica

HexLat[n_, finx_, finy_] := Module[{a = 0, L, R}, For[L = 0, L <= n, L++, For[R = Mod[n + finy - L, 2], R <= n - L , R += 2, a = a + Binomial[n, L]*Binomial[n - L, R]*Binomial[n - L - R, n/2 + L/2 - 3*R/2 + finx]*Binomial[n - L - R, (n - L - R - finy)/2]]]; a];
Table[HexLat[n, 2, 0], {n, 2, 25}] (* Jean-François Alcover, Jun 25 2023, after R. J. Mathar in A337905 *)

Formula

D-finite with recurrence (n-2)*(3*n^2-5*n-20)*(n+2)^2*a(n) -n*(3*n^4-2*n^3+n^2-130*n-208)*a(n-1) -24*n*(n-1)*(n-3)*(3*n^2+7*n-2)*a(n-2) -36*n*(n-1)*(n-2)*(3*n^2+n-22)*a(n-3)=0.

a(n) ~ 2^(n-1) * 3^(n + 1/2) / (Pi*n). - Vaclav Kotesovec, Apr 30 2024