A266549 -id:A266549 - cp's OEIS frontend

A000104 Number of n-celled free polyominoes without holes.

1, 1, 1, 2, 5, 12, 35, 107, 363, 1248, 4460, 16094, 58937, 217117, 805475, 3001127, 11230003, 42161529, 158781106, 599563893, 2269506062, 8609442688, 32725637373, 124621833354, 475368834568, 1816103345752, 6948228104703, 26618671505989, 102102788362303

Offset: 0

N. J. A. Sloane

J. S. Madachy, Pentominoes - Some Solved and Unsolved Problems, J. Rec. Math., 2 (1969), 181-188.
George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Mason, Table of n, a(n) for n = 0..40
Elena V. Konstantinova and Maxim V. Vidyuk, Discriminating tests of information and topological indices. Animals and trees, J. Chem. Inf. Comput. Sci. 43 (2003), 1860-1871.
John Mason, Description of counting programs
John Mason, Programs for calculation of numbers of unholey polyominoes
Lucia Moura and Ivan Stojmenovic, Backtracking and Isomorph-Free Generation of Polyhexes, Table 2.2 on p. 55 of Handbook of Applied Algorithms (2008).
W. R. Muller, K. Szymanski, J. V. Knop, and N. Trinajstic, On the number of square-cell configurations, Theor. Chim. Acta 86 (1993) 269-278
Tomás Oliveira e Silva, Enumeration of polyominoes
T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy)
R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Cf. A000105, row sums of A308300, A006746, A056877, A006748, A056878, A006747, A006749, A054361, A070765 (polyiamonds), A018190 (polyhexes), A266549 (by perimeter).

Cf. A056879, A056880, A056881, A056882, A006724, A357647, A357648.

a(n) = A000105(n) - A001419(n). - John Mason, Sep 06 2022

a(n) = (4*A056879(n) + 4*A056881(n) + 4*A056883(n) + 6*A056880(n) + 6*A056882(n) + 6*A357647(n) + 7*A357648(n) + A006724(n)) / 8. - John Mason, Oct 10 2022

Extended to n=26 by Tomás Oliveira e Silva

a(27)-a(28) from Tomás Oliveira e Silva's page added by Andrey Zabolotskiy, Oct 02 2022

A037245 Number of unrooted self-avoiding walks of n steps on square lattice.

Original entry on oeis.org

1, 2, 4, 9, 22, 56, 147, 388, 1047, 2806, 7600, 20437, 55313, 148752, 401629, 1078746, 2905751, 7793632, 20949045, 56112530, 150561752, 402802376, 1079193821, 2884195424, 7717665979, 20607171273, 55082560423, 146961482787, 392462843329, 1046373230168, 2792115083878

Offset: 1

Brian Hayes

Or, number of 2-sided polyedges with n cells. - Ed Pegg Jr, May 13 2009
A walk and its reflection (i.e., exchange start and end of walk, what Hayes calls a "retroreflection") are considered one and the same here. - Joerg Arndt, Jan 26 2018
With A001411 as main input and counting the symmetrical shapes separately, higher terms can be computed efficiently (see formula). - Bert Dobbelaere, Jan 07 2019

Bert Dobbelaere, Table of n, a(n) for n = 1..60
Joerg Arndt, The a(6) = 56 walks of length 6, 2018 (pdf, 2 pages).
Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyedge

Asymptotically approaches (1/16) * A001411.
Cf. A266549 (closed self-avoiding walks).
Cf. A323188, A323189 (program).

a(n) = (A001411(n) + A323188(n) + A323189(n) + 4) / 16. - Bert Dobbelaere, Jan 07 2019

a(25)-a(27) from Luca Petrone, Dec 20 2015

More terms using formula by Bert Dobbelaere, Jan 07 2019

A057730 Number of polyominoes (A000105) with perimeter 2n.

Original entry on oeis.org

0, 1, 1, 3, 6, 25, 86, 416, 1988, 10640, 57987, 328956, 1900321, 11204350, 67042778, 406780346

Offset: 1

N. J. A. Sloane, Oct 29 2000

Does this include polyominoes with holes? - Franklin T. Adams-Watters, Sep 12 2006. Answer from R. J. Mathar: Yes! See the illustrations in the links (e.g. perimeter 16, area 7, No 81 or perimeter 16, area 8, No 174).

All lines (sides of cells which are not common to a pair of cells) contribute to the perimeter, including the interior sides of cavities and holes. - R. J. Mathar, Feb 19 2021

Andrew Clarke, Isoperimetrical Polyominoes
Andrew Clarke, Picture from Andrew Clarke's page showing the polyominoes of perimeters 4, 6, 8 and 10.
Gordon Hamilton, Grade 3 area and perimeter activity
R. J. Mathar, perimeter 10
R. J. Mathar, perimeter 12
R. J. Mathar, perimeter 14
R. J. Mathar, Illustrating perimeter 16, area <= 13
R. J. Mathar, Illustrating perimeter 18, area <= 13
R. J. Mathar, Illustrating perimeter 20, area <= 13

Cf. A000105, A002931, A057753, A266549 (same, but holes not allowed), column sums of A342243, A131487 (polyominoes by total number of edges).

Additional comments from Barry Cipra, Jun 08 2004
Link updated by William Rex Marshall, Dec 16 2009
a(9)-a(10) added by Luca Petrone, Jan 08 2016
a(1)-a(9) confirmed by Bert Dobbelaere, Oct 19 2018
a(10)-a(12) corrected and extended by John Mason, Jul 26 2021
a(13)-a(16) added by John Mason, Sep 08 2022

A284869 Number of n-step 2-dimensional closed self-avoiding paths on triangular lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 5, 16, 37, 120, 344, 1175, 3807, 13224, 45645, 161705, 575325, 2074088, 7521818, 27502445, 101134999, 374128188

Offset: 1

Luca Petrone, Apr 04 2017

Differs from A057729 beginning at n = 11, since that sequence includes triangular polyominoes with holes.

a(n) is the number of simply connected polyiamonds with perimeter n. - Walter Trump, Nov 29 2023

Rade Doroslovački, Ivan Stojmenović and Ratko Tošić, Generating and counting triangular systems, BIT Numerical Mathematics, 27 (1987), 18-24. See Table 1.
Hugo Pfoertner, Illustration of ratio A036418(n)/a(n) using Plot2.
Hugo Pfoertner, Illustration of polygons of perimeter <= 11.
Walter Trump, Self-avoiding closed walks on a triangular lattice

Approaches (1/12)*A036418 for increasing n.

Cf. A057729, A258206, A266549, A316196.

a(15) from Hugo Pfoertner, Jun 27 2018

a(16)-a(22) from Walter Trump, Nov 29 2023

A316195 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/5, +-3Pi/5, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 2, 1, 18, 45, 441

Offset: 1

Hugo Pfoertner, Jun 26 2018

Holes are excluded, i.e., the boundary path may nowhere touch or intersect itself.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of polygons of perimeter <= 14.

Cf. A306175, A258206, A266549, A284869.

A316194 Number of symmetric self-avoiding polygons on square lattice with perimeter 2*n, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 1, 1, 3, 4, 16, 23, 87

Offset: 1

Hugo Pfoertner, Jun 27 2018

The sequence includes polygons of 2-fold, i.e., mirror or rotational, and higher (order >= 4) symmetry.

Hugo Pfoertner, Illustration of polygons of perimeter <= 16.

Cf. A002931, A266549, A316196, A323188, A323189.

A316197 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/7, +-3Pi/7, +-5*Pi/7, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 3, 4, 83, 533, 8329

Offset: 1

Hugo Pfoertner, Jun 28 2018

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of polygons of perimeter <= 12.

Cf. A306177, A258206, A266549, A284869, A316195.

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.

A316192 Number of self-avoiding polygons with perimeter n and sides = 1 that have vertex angles from the set 0, +-Pi/6, +-Pi/3, +-Pi/2, +-2Pi/3, +-5*Pi/6, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 1, 3, 4, 22, 69, 418, 2210, 14024

Offset: 1

Hugo Pfoertner, Jul 07 2018

Holes are excluded, i.e., the boundary path may nowhere touch or intersect itself.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of polygons of perimeter <= 8.

Cf. A258206, A266549, A284869, A306182, A316195, A316197, A316198, A316199, A316200, A316201.

A316200 Number of self-avoiding polygons with perimeter n and sides = 1 that have vertex angles from the set 0, +-Pi/5, +-2Pi/5, +-3Pi/5, +-4*Pi/5, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 0, 2, 2, 10, 15, 124, 352, 2378, 19405

Offset: 1

Hugo Pfoertner, Jul 07 2018

Holes are excluded, i.e., the boundary path may nowhere touch or intersect itself.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of polygons of perimeter <= 9.

Cf. A258206, A266549, A284869, A306180, A316192, A316195, A316197, A316198, A316199, A316201.

cp's OEIS Frontend

A000104 Number of n-celled free polyominoes without holes.

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A037245 Number of unrooted self-avoiding walks of n steps on square lattice.

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A057730 Number of polyominoes (A000105) with perimeter 2n.

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A284869 Number of n-step 2-dimensional closed self-avoiding paths on triangular lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

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A316195 Number of self-avoiding polygons with perimeter 2*n and sides = 1 that have vertex angles from the set +-Pi/5, +-3*Pi/5, not counting rotations and reflections as distinct.

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A316194 Number of symmetric self-avoiding polygons on square lattice with perimeter 2*n, not counting rotations and reflections as distinct.

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A316197 Number of self-avoiding polygons with perimeter 2*n and sides = 1 that have vertex angles from the set +-Pi/7, +-3*Pi/7, +-5*Pi/7, not counting rotations and reflections as distinct.

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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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A316192 Number of self-avoiding polygons with perimeter n and sides = 1 that have vertex angles from the set 0, +-Pi/6, +-*Pi/3, +-Pi/2, +-2*Pi/3, +-5*Pi/6, not counting rotations and reflections as distinct.

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A316200 Number of self-avoiding polygons with perimeter n and sides = 1 that have vertex angles from the set 0, +-Pi/5, +-2*Pi/5, +-3*Pi/5, +-4*Pi/5, not counting rotations and reflections as distinct.

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A316195 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/5, +-3Pi/5, not counting rotations and reflections as distinct.

A316197 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/7, +-3Pi/7, +-5*Pi/7, not counting rotations and reflections as distinct.

A316192 Number of self-avoiding polygons with perimeter n and sides = 1 that have vertex angles from the set 0, +-Pi/6, +-Pi/3, +-Pi/2, +-2Pi/3, +-5*Pi/6, not counting rotations and reflections as distinct.

A316200 Number of self-avoiding polygons with perimeter n and sides = 1 that have vertex angles from the set 0, +-Pi/5, +-2Pi/5, +-3Pi/5, +-4*Pi/5, not counting rotations and reflections as distinct.