A037245 -id:A037245 - cp's OEIS frontend

A266549 Number of 2n-step 2-dimensional closed self-avoiding paths on square lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

0, 1, 1, 3, 6, 25, 86, 414, 1975, 10479, 56572, 316577, 1800363, 10419605, 61061169, 361978851

Offset: 1

Luca Petrone, Dec 31 2015

Differs from A057730 beginning at n = 8, since that sequence includes polyominoes with holes.

Joerg Arndt, All a(6)=25 walks of length 12, 2018
Brendan Owen, Isoperimetrical Polyominoes, part of Andrew I. Clarke's Poly Pages.
Hugo Pfoertner, Illustration of ratio A002931(n)/a(n) using Plot2, showing apparent limit of 8.
Hugo Pfoertner, Illustration of polygons of perimeter <= 16.

Apparently lim A002931(n)/a(n) = 8 for increasing n, accounting for (in most cases) 4 rotations times two flips. - Joerg Arndt, Hugo Pfoertner, Jul 09 2018

Cf. A010566, A037245 (open self-avoiding walks), A316194.

a(11)-a(16) from Joerg Arndt, Jan 25 2018

A151541 Number of 2-sided triangular strip polyedges with n cells.

Original entry on oeis.org

1, 3, 8, 32, 123, 523, 2201, 9443, 40341, 172649, 736926, 3141607, 13367012, 56790498, 240919918, 1020753475, 4319803799, 18262494912, 77134873774, 325518862387, 1372679840360, 5784417772262

Offset: 1

Ed Pegg Jr, May 13 2009

Also number of unrooted self-avoiding walks of n steps on hexagonal [ =triangular ] lattice. - Hugo Pfoertner, Jun 23 2018

Ed Pegg, Jr., Illustrations of polyforms
Hugo Pfoertner, Illustration of ratio A001334(n)/a(n) using Plot2.
Eric Weisstein's World of Mathematics, Polyedge

Cf. A037245, A151539, A151540, A159867, A266925.

Asymptotically approaches (1/24)*A001334(n) for increasing n.

a(9)-a(13) from Joseph Myers, Oct 05 2011

a(14)-a(22) from Bert Dobbelaere, Mar 23 2025

A306178 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/4, +-Pi/2, +-3*Pi/4.

Original entry on oeis.org

1, 4, 15, 86, 492, 2992, 18172, 110643, 672267, 4069122, 24578785, 147972210, 889332713, 5331980703, 31924424199

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular octagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306179, A306180, A306181, A306182.

a(7)-a(15) from Bert Dobbelaere, May 15 2025

A306175 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/5, +-3*Pi/5.

Original entry on oeis.org

1, 2, 6, 19, 66, 229, 831, 2991, 10859, 39173, 141631, 510079, 1835583, 6586932, 23614821, 84492315, 302014619, 1077860479, 3843695976, 13690026688

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular pentagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 5.

Cf. A266925, A037245, A151541, A306177, A306178, A306179, A306180, A306181, A306182, A316195.

a(7)-a(9) from Hugo Pfoertner, Dec 23 2018

a(10)-a(20) from Bert Dobbelaere, May 15 2025

A306177 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/7, +-3Pi/7, +-5Pi/7.

Original entry on oeis.org

1, 3, 11, 55, 275, 1444, 7609, 40220, 212051, 1114206, 5840287, 30521627, 159166728, 828130291, 4301636648

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular heptagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 4.

Cf. A266925, A037245, A306175, A151541, A306178, A306179, A306180, A306181, A306182.

a(7)-a(15) from Bert Dobbelaere, May 15 2025

A306179 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/9, +-3Pi/9, +-5Pi/9, +-7*Pi/9.

Original entry on oeis.org

1, 4, 19, 128, 861, 6051, 42475, 297993, 2081607, 14485077, 100475175, 694838429, 4793805002

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular nonagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306180, A306181, A306182.

a(7)-a(13) from Bert Dobbelaere, May 15 2025

A306180 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/5, +-2Pi/5, +-3Pi/5, +-4*Pi/5.

Original entry on oeis.org

1, 5, 23, 169, 1233, 9551

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular decagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306179, A306181, A306182.

A306181 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/11, +-3Pi/11, +-5Pi/11, +-7Pi/11, +-9Pi/11.

Original entry on oeis.org

1, 5, 28, 235, 1970, 17201, 149420, 1295637, 11178026, 96047288, 822418731, 7020762655

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular 11-gon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306179, A306180, A306182.

a(7)-a(12) from Bert Dobbelaere, May 15 2025

A306182 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/6, +-2Pi/6, +-3Pi/6, +-4Pi/6, +-5Pi/6.

Original entry on oeis.org

1, 6, 33, 306, 2765, 26290, 247737, 2332965, 21856232, 204019196, 1897940592, 17606864337

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular 12-gon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306179, A306180, A306181.

a(7)-a(12) from Bert Dobbelaere, May 15 2025

A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 15, 16, 18, 20, 21, 23, 24, 25, 26, 27, 28, 32, 34, 36, 38, 44, 46, 48, 52, 56, 58, 60

Offset: 1

Hugo Pfoertner and Markus Sigg, Jul 30 2021

Closed walks are allowed.

			See link for illustrations of terms corresponding to diameters D < 8.5.

Hugo Pfoertner, Examples of paths of maximum length.

The squared radii of the enclosing circles are a subset of A192493/A192494.
Cf. A037245, A122224, A127399, A127400, A127401, A266549, A316194, A346993, A346994, A346995.
Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

cp's OEIS Frontend

A266549 Number of 2n-step 2-dimensional closed self-avoiding paths on square lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

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A151541 Number of 2-sided triangular strip polyedges with n cells.

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A306178 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/4, +-Pi/2, +-3*Pi/4.

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A306175 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/5, +-3*Pi/5.

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A306177 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/7, +-3*Pi/7, +-5*Pi/7.

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A306179 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/9, +-3*Pi/9, +-5*Pi/9, +-7*Pi/9.

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A306180 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/5, +-2*Pi/5, +-3*Pi/5, +-4*Pi/5.

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A306181 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/11, +-3*Pi/11, +-5*Pi/11, +-7*Pi/11, +-9*Pi/11.

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A306182 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/6, +-2*Pi/6, +-3*Pi/6, +-4*Pi/6, +-5*Pi/6.

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A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

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A306177 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/7, +-3Pi/7, +-5Pi/7.

A306179 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/9, +-3Pi/9, +-5Pi/9, +-7*Pi/9.

A306180 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/5, +-2Pi/5, +-3Pi/5, +-4*Pi/5.

A306181 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/11, +-3Pi/11, +-5Pi/11, +-7Pi/11, +-9Pi/11.

A306182 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/6, +-2Pi/6, +-3Pi/6, +-4Pi/6, +-5Pi/6.