A357771 Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-6 node.

1, 6, 60, 744, 10224, 148896, 2250816, 34922880, 552386304, 8867756544, 144044098560, 2362292213760, 39049785446400, 649843233546240, 10876273137008640, 182934715370471424, 3090181365862170624, 52398620697685524480, 891492911924665122816, 15213249205591283859456, 260315328935885892747264

Offset: 0

Author

Dave R.M. Langers, Oct 12 2022

Keywords

• nonn
• easy
• walk

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-6 nodes alternate with degree-3 nodes.

Examples

			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.
		

Crossrefs

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.

Programs

Mathematica

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

Formula

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

Extensions

More terms from Detlef Meya, May 15 2024