A019988 -id:A019988 - cp's OEIS frontend

A151538 Number of 1-sided strip polyedges with n cells.

1, 2, 6, 14, 40, 102, 284, 752, 2069, 5547, 15134, 40712, 110456, 297066, 802808, 2156378, 5810329, 15584271, 41894990, 112217372, 301115391, 805584175, 2158366236, 5768337730, 15435275815, 41214200699, 110164972820, 293922598172, 784925297952, 2092745480990, 5584229143243

Offset: 1

Ed Pegg Jr, May 13 2009

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
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyedge

Cf. A001411, A019988, A037245, A151537, A323189 (program).

a(n) = (A001411(n) + A323189(n)) / 8. - Bert Dobbelaere, Jan 07 2019

a(13)-a(19) from Joseph Myers, Oct 03 2011

More terms using formula by Bert Dobbelaere, Jan 07 2019

A383981 Number of connected subsets of n edges of the rhombic dodecahedron up to the 48 rotations and reflections of the rhombic dodecahedron.

Original entry on oeis.org

1, 1, 3, 5, 16, 39, 127, 357, 1067, 2861, 7071, 14827, 25638, 33730, 33189, 24838, 14954, 7188, 2905, 912, 254, 49, 11, 1, 1

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 48 symmetries of the rhombic dodecahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383982 Number of connected subsets of n edges of the cuboctahedron up to the 48 rotations and reflections of the cuboctahedron.

Original entry on oeis.org

1, 1, 3, 7, 24, 74, 269, 876, 2788, 7639, 17828, 32326, 44375, 46456, 39213, 26865, 15470, 7278, 2917, 913, 254, 49, 11, 1, 1

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 48 symmetries of the cuboctahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383983 Number of connected subsets of n edges of the rhombic triacontahedron up to the 120 rotations and reflections of the rhombic triacontahedron.

Original entry on oeis.org

1, 1, 3, 7, 24, 84, 334, 1330, 5495, 22776, 94920, 394706

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 120 symmetries of the rhombic triacontahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383984 Number of connected subsets of n edges of the icosidodecahedron up to the 120 rotations and reflections of the icosidodecahedron.

Original entry on oeis.org

1, 1, 3, 7, 24, 81, 323, 1265, 5202, 21335, 88412, 364897

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 120 symmetries of the icosidodecahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A151537 Number of 1-sided polyedges with n edges.

Original entry on oeis.org

1, 2, 7, 25, 99, 416, 1854, 8411, 38980, 182829, 867096, 4145168, 19955321, 96619260, 470157772

Offset: 1

Ed Pegg Jr, May 13 2009

Ishino Keiichiro, Polyedge, Puzzle will be played, 2009.
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyedge

Cf. A019988, A096267, A037245, A151538, A333249.

a(11)-a(14) from Joseph Myers, Oct 03 2011

a(15) from Ishino Keiichiro's website added by Andrey Zabolotskiy, Dec 10 2023

A348095 Number of free n-polysticks embedded in the square lattice with at least one cycle.

Original entry on oeis.org

0, 0, 0, 1, 1, 10, 42, 254, 1331, 7358, 39543, 212911, 1135876, 6039878, 31975124, 168790048, 888664299, 4669177072

Offset: 1

R. J. Mathar, Sep 30 2021

The number of holes h = e - v + 1 (e = the number of edges and v the number of vertices). - John Mason, Feb 12 2023

			The 4-stick with a cycle is the unit square. The 5-stick with a cycle is the unit square with one protruding edge. The 10 6-sticks with a cycle are the unit square with two protruding edges (in various cis, meta, trans configurations) or with a protruding 2-stick at various angles, or a 2x1 rectangle.
Size 6 examples 1 through 5:
  +-+        +-+    +        +    +-+-+
  | |        | |    |        |      | |
  +-+-+-+  +-+-+-+  +-+-+  +-+-+    +-+-+
                    | |    | |
                    +-+    +-+
Examples 6 through 10:
  +        +-+ +    +      +-+-+  +-+-+
  |        | | |    |      | |    |   |
  +-+      +-+-+    +-+-+  +-+-+  +-+-+
  | |                 | |
  +-+-+               +-+

John Mason, Examples of size 6

Cf. A056841 (tree-like), A019988 (free polysticks).

a(n) + A056841(n+1) = A019988(n).

a(14)-a(18) from John Mason, Jun 01 2023

A361625 Number of free polyominoes with checkerboard-pattern-colored vertices with n cells.

Original entry on oeis.org

1, 1, 3, 7, 20, 60, 204, 702, 2526, 9180, 33989, 126713, 476597, 1802109, 6850969, 26151529, 100207548, 385217382, 1485216987, 5741240989, 22246000726, 86383317470, 336093551268, 1309997856337, 5114452295933, 19998171631076, 78306014924606, 307022177714062

Offset: 1

Andrey Zabolotskiy, Mar 19 2023; thanks to John Mason for his help

nonn

Also, number of polysticks of size n (see A019988), with the requirement that any two sticks are connected by a sequence of adjacent, alternately horizontal and vertical sticks. - Pontus von Brömssen, Sep 01 2023

			There are 2 ways to color the 4 corners of a monomino with black and white colors alternatingly, but they are related by a rotation or a reflection, so a(1) = 1. a(2) is also 1 because the two ways to color the 6 vertices of a domino with black and white colors in the checkerboard pattern are related to each other by a reflection or a rotation. The same is true for the stick tromino, but the two ways to color the 8 vertices of the L-tromino are inequivalent, so a(3) = 3.
For n = 3, the a(3) = 3 allowed polysticks are:
  _     _
  _|  _|   _|_

Andrey Zabolotskiy, Table of n, a(n) for n = 1..50
Index entries for sequences related to polyominoes.

A122675 is the 3-dimensional analog based on polycubes.
Cf. A001933, A019988, A359689.
Cf. A000105, A351190, A351142, A351127, A349328, A346799, A234008.
5th row of A366766.

a(n) = 2 * A000105(n) - (A351190(n) + A351142(n) + A351127(n) + A349328(n) + A346799(n/2) + A234008(n/2)), where the last two terms are only included if 2|n. I.e., every free polyomino is counted twice here unless it is symmetric with respect to a Pi/2 rotation centered at a cell, or a Pi rotation centered at an edge, or a reflection with respect to an axis parallel to the grid and passing through cells.

A385390 Irregular triangle read by rows: T(n,k) is the number of polysticks of size k, i.e., connected subsets of k edges, of the n X n flat torus, up to cyclic shifts and reflections of rows and columns, as well as interchange of rows and columns; 1 <= k <= 2*n^2.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 4, 4, 1, 1, 1, 2, 5, 14, 38, 111, 261, 500, 654, 648, 486, 305, 144, 61, 19, 6, 1, 1, 1, 2, 5, 16, 52, 199, 759, 2921, 10668, 36761, 115231, 322237, 778242, 1576259, 2591721, 3412285, 3671098, 3320276, 2565917, 1717088, 996355, 503860, 220074, 83408, 26783, 7438, 1678, 351, 52, 11, 1, 1

Offset: 1

Pontus von Brömssen, Jun 27 2025

For n = 4, there are 384 automorphisms of (the line graph of) the 4 X 4 torus grid graph (it is isomorphic to the 4-dimensional hypercube graph), but here we only consider the subgroup consisting of the 128 symmetries of the 4 X 4 torus. Using the full automorphism group of the torus grid graph would change row 4 to the corresponding row of A333333.

			Triangle begins:
  1, 1;
  1, 2, 3,  7,  4,   4,   1,   1;
  1, 2, 5, 14, 38, 111, 261, 500, 654, 648, 486, 305, 144, 61, 19, 6, 1, 1;
  ...

Cf. A019988, A333333, A385385 (polyominoes), A385388 (interchange of rows and columns of the torus not allowed), A385389 (row sums).

T(n,k) = A019988(k) if n >= k.

T(n,k) >= A385388(n,k)/2, with equality if and only if k is odd.

A385583 Triangle read by rows: T(n,d) is the number of free d-dimensional polysticks of size n.

Original entry on oeis.org

1, 1, 2, 1, 5, 7, 1, 16, 28, 31, 1, 55, 160, 199, 205, 1, 222, 1085, 1651, 1768, 1779

Offset: 1

Pontus von Brömssen, Jul 04 2025

If d > n, there are T(n,n) such polysticks. The triangle only includes the values for d <= n.

			Triangle begins:
  n\d| 1   2    3    4    5    6
  ---+--------------------------
  1  | 1
  2  | 1   2
  3  | 1   5    7
  4  | 1  16   28   31
  5  | 1  55  160  199  205
  6  | 1 222 1085 1651 1768 1779

Index entries for sequences related to polyominoes.

Cf. A330891 (polyominoes), A365565 (main diagonal), A365566, A385581 (fixed).

Columns: A019988 (d=2), A365559 (d=3), A365561 (d=4), A365563 (d=5).

T(n,d) = Sum_{k=1..d} A365566(n,k).

cp's OEIS Frontend

A151538 Number of 1-sided strip polyedges with n cells.

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A383981 Number of connected subsets of n edges of the rhombic dodecahedron up to the 48 rotations and reflections of the rhombic dodecahedron.

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A383982 Number of connected subsets of n edges of the cuboctahedron up to the 48 rotations and reflections of the cuboctahedron.

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A383983 Number of connected subsets of n edges of the rhombic triacontahedron up to the 120 rotations and reflections of the rhombic triacontahedron.

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A383984 Number of connected subsets of n edges of the icosidodecahedron up to the 120 rotations and reflections of the icosidodecahedron.

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A151537 Number of 1-sided polyedges with n edges.

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A348095 Number of free n-polysticks embedded in the square lattice with at least one cycle.

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A361625 Number of free polyominoes with checkerboard-pattern-colored vertices with n cells.

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A385390 Irregular triangle read by rows: T(n,k) is the number of polysticks of size k, i.e., connected subsets of k edges, of the n X n flat torus, up to cyclic shifts and reflections of rows and columns, as well as interchange of rows and columns; 1 <= k <= 2*n^2.

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A385583 Triangle read by rows: T(n,d) is the number of free d-dimensional polysticks of size n.

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