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The number of walks of n steps on the hexagonal lattice that start at the origin and end at the non-adjacent vertex (3/2,sqrt(3)/2).
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%I #6 Oct 01 2020 07:42:45

%S 2,6,48,220,1320,6930,39200,215208,1208340,6754440,38076192,214939296,

%T 1218641424,6925848930,39477746880,225542306704,1291514481972,

%U 7410367503396,42599109627360,245305128355560,1414839151645920,8172376003368720,47270088643265280,273766119948648000

%N The number of walks of n steps on the hexagonal lattice that start at the origin and end at the non-adjacent vertex (3/2,sqrt(3)/2).

%F D-finite with recurrence -(n-2)*(n+3)*(n+2)*(n+1)*a(n) +n*(n+2)*(n^2+n+12)*a(n-1) +24*n*(n-1)*(n^2+3*n-1)*a(n-2) +36*n*(n-1)*(n-2)*(n+4)*a(n-3)=0.

%e There are a(2)=2 paths with 2 steps: RU and UR, 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: UUD, UDU, DUU, RRu, RuR, uRR.

%p # see A337905

%Y Cf. A002898 (returns to origin), A337905, A337906.

%K nonn,walk

%O 2,1

%A _R. J. Mathar_, Sep 29 2020