A334720 - OEIS

A334720

Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps with incrementing length from 1 to n.

0, 0, 0, 0, 0, 0, 8, 24, 0, 0, 40, 112, 0, 0, 1376, 2008, 0, 0, 21720, 60848, 0, 0, 635544, 1517368, 0, 0, 20008456, 46010640, 0, 0, 640819936, 1571759136, 0, 0, 22704325648, 55436103264

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OFFSET

1,7

COMMENTS

This sequence gives the number of closed-loop self avoiding walks on a 2D square lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. No closed-loop path is possible until n = 7.

Like A010566 all possible paths are counted, including those that are equivalent via rotation and reflection.

LINKS

Table of n, a(n) for n=1..36.

A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice, J. Phys. A 21 (1988), L165-L172.

Scott R. Shannon, Images of the closed-loops for n=7,8,11,12,15.

EXAMPLE

a(1) to a(6) = 0 as no closed-loop is possible.

a(7) = 8 as there is one path which forms a closed loop which can be walked in 8 different ways on a 2D square lattice. The path is:

*---.---.---.---.---*

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