0,2

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.

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

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.

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 *)

(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

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

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

Sequence in context: A112117 A065944 A393380 * A392645 A126779 A218441

Adjacent sequences: A357768 A357769 A357770 * A357772 A357773 A357774

nonn,easy,walk

Dave R.M. Langers, Oct 12 2022

More terms from Detlef Meya, May 15 2024

approved