OFFSET
0,2
COMMENTS
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-3 nodes alternate with degree-6 nodes.
FORMULA
a(0) = 1; a(n) = Sum_{k=0..n} (binomial(n, k) * Sum_{j=0..n} (binomial(n, j) * Sum_{i= 0..j} ((1/(2^(j + 1)))*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i))). - Detlef Meya, May 20 2024
EXAMPLE
a(2)=30, because there are 3*3=9 paths that visit one of three adjacent vertices, return to the origin, and again visit an adjacent vertex and return to the origin; 3*5=15 paths visiting one of five distance-2 vertices that are adjacent to the three adjacent vertices; plus 3*2=6 paths traversing the perimeter of three adjacent rhombi in counterclockwise or clockwise direction; all resulting in a closed path of length 2n=2*2=4.
MATHEMATICA
a[0] := 1; a[n_] := Sum[Binomial[n, k]*Sum[Binomial[n, j]*Sum[(1/(2^(j+1)))*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, 19}]] (* Detlef Meya, May 20 2024 *)
CROSSREFS
The accompanying sequences for the number of paths that return to a degree-6 node is A357771.
KEYWORD
nonn,easy,walk,more
AUTHOR
Dave R.M. Langers, Oct 12 2022
EXTENSIONS
More terms from Detlef Meya, May 20 2024
STATUS
approved