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

2,2

Links

Table of n, a(n) for n=2..25.

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A370224 A218990 A087413 * A244829 A059228 A079568

Adjacent sequences: A337903 A337904 A337905 * A337907 A337908 A337909

Keyword

nonn,walk

Author

R. J. Mathar, Sep 29 2020

Status

approved