A266549 -id:A266549 - cp's OEIS frontend

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

A342536 Number of self-avoiding polygons on the square lattice, of perimeter 2n, with the property that all the right-angles of the same orientation are contiguous.

Original entry on oeis.org

1, 1, 3, 4, 10, 17, 36, 65, 126, 227, 419, 743, 1323, 2295, 3965

Offset: 2

John Mason, Allan C. Wechsler, Mar 15 2021

For any polygon, build a bracelet of black and white beads by following the border of the polygon in a clockwise direction, adding a black bead for each right-turning right angle, and a white bead for each left-turning right angle. The polygons counted by this sequence are those whose bracelets have all the black beads together and all the white beads together.

			a(4)=3, as there are 3 self-avoiding polygons (SAPs) of perimeter 8 that satisfy the condition; these are the polygons corresponding to the strip and L-shaped trominoes, and the square tetromino.

John Mason, Contiguously oriented saps

Cf. A266549.

A361288 Number of free polyominoes of size 2n for which there exists at least one closed path that passes through each square exactly once.

Original entry on oeis.org

1, 1, 3, 6, 25, 84, 397, 1855, 9708, 51684, 286011, 1609097, 9222409, 53543338, 314612803

Offset: 2

John Mason and Tanya Khovanova, Mar 07 2023

A polyomino for which more than one closed path exists counts as 1. On the other hand, in A266549, distinct closed paths count separately. For example for n=7, this latter sequence distinguishes between
     +-+ +-+
     | | | |
     + +-+ +-+
     |       |
     +-+-+-+-+
and
     +-+-+-+
     |     |
     + +-+ +-+
     | | |   |
     +-+ +-+-+

			For n = 4 the a(4) = 3 solutions are:
  XXX    XX   XXXX
  X X   XXX   XXXX
  XXX   XXX

John Mason, Examples

Cf. A266549 (where distinct closed paths count separately).

a(13) - a(16) from Bert Dobbelaere, Mar 09 2023

A320421 Number of closed self-avoiding paths on square lattice with 2*n steps, A010566(n)/8.

Original entry on oeis.org

0, 1, 3, 14, 70, 372, 2058, 11752, 68706, 409130, 2472646, 15127620, 93504944, 583032968, 3662883960, 23163461280, 147329432094, 941880843108, 6049001532148, 39007700026460, 252477751201074, 1639657957610596, 10680997864879592, 69772819359471480, 456959583009324200

Offset: 1

Hugo Pfoertner, Jan 08 2019

Rotations and reflections of the path are counted only once, but see A266549 for further elimination of path isomorphisms.

See A010566.

Cf. A002931, A010566, A266549.

cp's OEIS Frontend

A316201 Number of self-avoiding polygons with perimeter 2*n and sides = 1 that have vertex angles from the set +-Pi/11, +-3*Pi/11, +-5*Pi/11, +-7*Pi/11, +-9*Pi/11, not counting rotations and reflections as distinct.

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A323134 Number of polygons made of uncrossed knight's paths of length 2*n on an infinite board.

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

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A342536 Number of self-avoiding polygons on the square lattice, of perimeter 2n, with the property that all the right-angles of the same orientation are contiguous.

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A361288 Number of free polyominoes of size 2n for which there exists at least one closed path that passes through each square exactly once.

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Author

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Examples

Links

Crossrefs

Extensions

A320421 Number of closed self-avoiding paths on square lattice with 2*n steps, A010566(n)/8.

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Author

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A316201 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/11, +-3Pi/11, +-5Pi/11, +-7Pi/11, +-9*Pi/11, not counting rotations and reflections as distinct.