A030222 -id:A030222 - cp's OEIS frontend

A239658 Number of free triangular n-polyplets (triangular polyominoes connected at edges or corners), where turning over is allowed and holes are allowed.

1, 1, 3, 11, 75, 528, 4584, 40609, 373981, 3493723, 33079702, 316109657, 3044648053

Offset: 0

Abe Wits and Ragnar Groot Koerkamp, Nov 20 2020

We get triangular polyominoes when connecting over edges only (rather than edges and corners); that is A000577. The square equivalent (that is, n-polyplets) is A030222. The neighborhood is also known as the Triangular Moore neighborhood, and equals the moves of the Tri-Chess king.

Cf. A000577, A030222.

a(11) and a(12) from Aaron N. Siegel, May 22 2022

A255540 Number of connected digital images in Z^2 (regarding each point as connected to its 8 neighbors) on n points up to isomorphism.

Original entry on oeis.org

1, 1, 2, 6, 15, 51, 173, 681, 2682

Offset: 1

P. Christopher Staecker, Feb 24 2015

Also the number of polyplets (A030222) on n points up to isomorphism of adjacency graphs.

			For n=4 the a(4)=6 images are:
X   X X    X    XX    X     X
X    X    X X   XX   XX   XX
X    X     X         X     X
X

P. C. Staecker, Some enumerations of binary digital images, arXiv 1502.06236 [math.CO], 2015 (see table p. 6).
P. C. Staecker, Digital image enumerations

Cf. A255539.

A364927 List of free polyplets in binary code (as defined in A246521), ordered first by number of bits, then by value of the binary code.

Original entry on oeis.org

1, 3, 6, 7, 11, 14, 25, 56, 15, 23, 27, 29, 30, 46, 57, 58, 75, 78, 89, 92, 120, 166, 177, 178, 198, 209, 240, 390, 452, 960, 31, 47, 59, 62, 79, 91, 93, 94, 110, 121, 122, 124, 143, 167, 174, 179, 181, 182, 185, 186, 188, 199, 206, 211, 213, 230, 241, 242

Offset: 1

Pontus von Brömssen, Aug 13 2023

Can be read as an irregular triangle, whose n-th row contains A030222(n) terms.

			As irregular triangle:
  1;
  3,  6;
  7, 11, 14, 25, 56;
  ...
The A030222(3) = 5 3-polyplets are oriented as follows to obtain their binary codes (see A246521):
  . . .   . . .   . . .   . . .   5 . .
  2 . .   . . .   2 . .   . 4 .   . 4 .
  0 1 .   0 1 3   . 1 3   0 . 3   . . 3
This gives the binary codes 2^0+2^1+2^2 = 7, 2^0+2^1+2^3 = 11, 2^1+2^2+2^3 = 14, 2^0+2^3+2^4 = 25, and 2^3+2^4+2^5 = 56, respectively.

Pontus von Brömssen, Table of n, a(n) for n = 1..22449 (rows 1..8).

Cf. A030222, A246521.

A030233 Number of n-celled one-sided polyplets.

Original entry on oeis.org

1, 2, 6, 34, 166, 991, 5931, 37196, 235456, 1514618, 9826177, 64284947, 423241426, 2802300793, 18642694440, 124544085550, 835091956750

Offset: 1

David W. Wilson

Eric Weisstein's World of Mathematics, Polyplet.

Cf. A030222.

More terms from Joseph Myers, Sep 26 2002

A030234 Number of n-celled polyplets having bilateral symmetry.

Original entry on oeis.org

1, 2, 4, 10, 22, 57, 131, 344, 810, 2144, 5127, 13645, 32984, 88133, 214768, 575578, 1411442

Offset: 1

David W. Wilson

Eric Weisstein's World of Mathematics, Polyplet.

Cf. A030222.

More terms from Joseph Myers, Sep 26 2002

A030235 Number of n-celled polyplets without bilateral symmetry.

Original entry on oeis.org

0, 0, 1, 12, 72, 467, 2900, 18426, 117323, 756237, 4910525, 32135651, 211604221, 1401106330, 9321239836, 62271754986, 417545272654

Offset: 1

David W. Wilson

Eric Weisstein's World of Mathematics, Polyplet.

Cf. A030222.

More terms from Joseph Myers, Sep 26 2002

A162678 Number of fixed strictly disconnected n-ominoes bounded (not necessarily tightly) by an n*n square.

Original entry on oeis.org

0, 2, 42, 937, 26427, 937126, 40290848, 2036152559, 118202398712, 7747410863508, 565695467280668, 45525704815211707, 4002930269942820774, 381750656962679848234, 39244733577786597223238

Offset: 1

David Bevan, Jul 28 2009

nonn

a(n) = A162676(n) - A001168(n)

			a(2)=2: the two rotations of the disconnected domino consisting of two squares connected at a vertex

Cf. A162676, A001168, A006770, A030222

A194596 Number of free polyplets with n cells that are not polyominoes.

Original entry on oeis.org

0, 1, 3, 17, 82, 489, 2923, 18401, 116848, 753726, 4898579, 32085696, 211398614, 1400292492, 9318028028, 62259251309, 417496576187

Offset: 1

Jani Melik, Aug 30 2011

			XX...X.X..X.. (the 3 for n=3)
..X...X....X.
............X
a(3) = 5 - 2 = 3
a(4) = 22 - 5 = 17
a(5) = 94 - 12 = 82

Eric Weisstein's World of Mathematics, Polyplet.

Cf. A030222, A000105, A006770 (fixed polyplets), A001168 (fixed polyominoes).

a(n) = A030222(n) - A000105(n).

A356975 a(n) is the number of n-celled pseudo-polyominoes with holes.

Original entry on oeis.org

0, 0, 0, 1, 3, 28, 199, 1603, 12234, 93270, 697641, 5167477, 37913465

Offset: 1

John Mason, Sep 07 2022

			a(4)=1 because of:
   O
  O O
   O
a(5)=3 because of:
   O    O     O
  O O  O OO  O O
  OO    O     O O

Cf. A030222, A001419.

cp's OEIS Frontend

A239658 Number of free triangular n-polyplets (triangular polyominoes connected at edges or corners), where turning over is allowed and holes are allowed.

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A255540 Number of connected digital images in Z^2 (regarding each point as connected to its 8 neighbors) on n points up to isomorphism.

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A364927 List of free polyplets in binary code (as defined in A246521), ordered first by number of bits, then by value of the binary code.

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A030233 Number of n-celled one-sided polyplets.

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A030234 Number of n-celled polyplets having bilateral symmetry.

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A030235 Number of n-celled polyplets without bilateral symmetry.

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A162678 Number of fixed strictly disconnected n-ominoes bounded (not necessarily tightly) by an n*n square.

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A194596 Number of free polyplets with n cells that are not polyominoes.

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A356975 a(n) is the number of n-celled pseudo-polyominoes with holes.

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