A337550 -id:A337550 - cp's OEIS frontend

A156228 Number of lakes in Conway's Game of Life with 8*n cells.

1, 0, 1, 1, 4, 7, 31, 98, 446, 1894, 9049, 43151

Offset: 1

Nathaniel Johnston, Feb 06 2009

a(n) is also the number of walks of length 4*n on a 2D lattice with the properties that: it turns 90 degrees after every step of length 1, it is a closed loop (i.e., it ends where it started) and it never crosses itself.

In A266549, the walks are allowed to continue straight ahead. - Pontus von Brömssen, May 06 2025

			a(2) = 0 because there are no lakes with 16 cells.

Nathaniel Johnston, Counting Lakes
LifeWiki, Lakes

Cf. A019473, A266549, A337550.

a(11) and a(12) added by Nathaniel Johnston, Mar 09 2009

A345676 Number of closed-loop self-avoiding paths on a 2-dimensional square lattice where each path consists of steps with successive lengths equal to the square numbers, from 1 to n^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 368, 264, 0, 0, 1656, 5104, 0, 0, 62016, 105344, 0, 0, 1046656, 3181104

Offset: 1

Scott R. Shannon, Sep 04 2021

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 at each step to the next square number until the step length is n^2. No closed-loop path is possible until n = 15.
Like A334720 and A335305 the only n values that can form closed loop walks are those which correspond to the indices of even triangular numbers. Curiously though n = 16 walks form no closed loops, even though both n = 15 and n = 16 are indices of such numbers.
As in A010566 all possible paths are counted, including those that are equivalent via rotation and reflection.

			a(1) to a(14) = 0 as no closed-loop paths are possible.
a(15) = 32 as there are four different paths which form closed loops, and each of these can be walked in eight different ways on a 2D square lattice. These walks consist of steps with lengths 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. See the linked text images.

A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.
Scott R. Shannon, Images of the closed loops for n = 15. The line lengths in this text file are long so it may need to be downloaded to be viewed correctly.

Cf. A347506, A334720, A336265, A337550, A010566, A000217.

cp's OEIS Frontend

A156228 Number of lakes in Conway's Game of Life with 8*n cells.

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