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%I #10 Jun 25 2023 09:38:02
%S 1,6,34,200,1095,6230,34636,195552,1099140,6218520,35210736,200099328,
%T 1139401263,6504768270,37211008120,213311576192,1225053737622,
%U 7047867363108,40612374024300,234376628718960,1354498970905080,7838134441085520,45412879702456800,263417461793120000
%N 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).
%F 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.
%e 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).
%e There are a(3)=6 paths with 3 steps: RUD, RDU, DRU, DUR, URD, UDR.
%p # see A337905
%t 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];
%t Table[HexLat[n, 2, 0], {n, 2, 25}] (* _Jean-François Alcover_, Jun 25 2023, after _R. J. Mathar_ in A337905 *)
%Y Cf. A002898 (returns to origin), A337905, A337907.
%K nonn,walk
%O 2,2
%A _R. J. Mathar_, Sep 29 2020
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