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Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-6 node.
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%I #28 May 20 2024 11:04:42

%S 1,6,60,744,10224,148896,2250816,34922880,552386304,8867756544,

%T 144044098560,2362292213760,39049785446400,649843233546240,

%U 10876273137008640,182934715370471424,3090181365862170624,52398620697685524480,891492911924665122816,15213249205591283859456,260315328935885892747264

%N Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-6 node.

%C Paths that return to the same point in a quasi-regular rhombic lattice must always have even length (i.e., 2n) because of parity: degree-6 nodes alternate with degree-3 nodes.

%F a(n) = Sum_{k=0..n} (binomial(n, k) * Sum_{j=0..n} (binomial(n, j) * Sum_{i= 0..j} ((1/(2^j))*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i))). - _Detlef Meya_, May 15 2024

%e a(2)=60, because there are 6*6=36 paths that visit one of six adjacent vertices, return to the origin, and again visit an adjacent vertex and return to the origin; plus 6*4=24 paths that pass through one of the six vertices at distance 2, leaving and returning via any of two available paths to that vertex; all resulting in a closed path of length 2n=2*2=4.

%t a[n_] := Sum[Binomial[n, k]*Sum[Binomial[n, j]*Sum[(1/(2^j))*Binomial[2*i, j]*Binomial[2*i, i]*Binomial[2*(j-i), j-i],{i,0,j}],{j,0,n}],{k,0,n}]; Flatten[Table[a[n],{n,0,17}]] (* _Detlef Meya_, May 15 2024 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n, k) * sum(j=0, n, binomial(n, j) * sum(i=0, j, (1/(2^j)*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i))))); \\ _Michel Marcus_, May 20 2024

%Y The accompanying sequences for the number of paths that return to a degree-3 node is A357770.

%Y Similar sequences for square, hexagonal, and honeycomb lattices are A002894, A002898 and A002893.

%K nonn,easy,walk,more

%O 0,2

%A _Dave R.M. Langers_, Oct 12 2022

%E More terms from _Detlef Meya_, May 15 2024