A357770 Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-3 node.

1, 3, 30, 372, 5112, 74448, 1125408, 17461440, 276193152, 4433878272, 72022049280, 1181146106880, 19524892723200, 324921616773120

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.

Links

Table of n, a(n) for n=0..13.

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.

Crossrefs

The accompanying sequences for the number of paths that return to a degree-6 node is A357771.

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

Sequence in context: A178016 A365152 A372087 * A372105 A372091 A372110

Adjacent sequences: A357767 A357768 A357769 * A357771 A357772 A357773

Keyword

nonn,easy,walk,more

Author

Dave R.M. Langers, Oct 12 2022

Status

approved