A334398 - OEIS

A334398

Number of endless self-avoiding walks of length n for the square lattice up to rotation, reflection, and path reversal.

1, 2, 3, 7, 16, 39, 96, 245, 631, 1642, 4304, 11272, 29690, 78059, 205805, 542229, 1429948, 3770287, 9944412, 26227636, 69182561, 182488890, 481379260, 1269847657, 3349768007

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OFFSET

1,2

COMMENTS

a(n) > A334322(n)/16.

Does not reduce for repetition (e.g. paths XYXY and XXYY both count for n = 4, even though they are respectively a serial and an interleaved repetition of primitive path XY for n = 2), or for congruence of the resultant endless paths (e.g. paths XXYY and XYYX each count even though their resultant endless paths are congruent). - Charles L. Hohn, May 11 2026

LINKS

Table of n, a(n) for n=1..25.

Nathan Clisby, Endless self-avoiding walks, arXiv:1302.2796 [cond-mat.stat-mech], 2013.

Code Golf Stack Exchange, Infinite Snake game

EXAMPLE

The a(4) = 7 step patterns of length 4 are

---->---->---->---->, ---->---->---->, ---->----> ,

| |

v v---->

---->---->, ----> ^, ----> , and ----> .

| | | | |

v v----> v----> v