cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A056785 Number of polydominoes.

Original entry on oeis.org

1, 4, 23, 211, 2227, 25824, 310242, 3818983, 47752136, 604425323
Offset: 1

Views

Author

James Sellers, Aug 28 2000

Keywords

Comments

From Vicher's table.
This gives the number of polyominoes of order 2n that can be tiled by dominoes in at least one way. - Joseph Myers, Jun 10 2012

Links

Crossrefs

Cf. A056786, A213376, A213377, A213378.

Formula

a(n) + A213376(n) = A210996(n). - R. J. Mathar, Jan 30 2023

Extensions

Edited by T. D. Noe, Apr 09 2009
Offset corrected and a(6)-a(9) added by Joseph Myers, Jun 10 2012
a(10) added by Arthur O'Dwyer, Aug 28 2025

A210996 Number of free polyominoes with 2n cells.

Original entry on oeis.org

1, 1, 5, 35, 369, 4655, 63600, 901971, 13079255, 192622052, 2870671950, 43191857688, 654999700403, 9999088822075, 153511100594603, 2368347037571252, 36695016991712879, 570694242129491412, 8905339105809603405, 139377733711832678648, 2187263896664830239467, 34408176607279501779592
Offset: 0

Views

Author

Omar E. Pol, Sep 15 2012

Keywords

Comments

It appears that we can write A216492(n) < A216583(n) < A056785(n) < A056786(n) < a(n) < A210988(n) < A210986(n), if n >= 3. - Omar E. Pol, Sep 16 2012

Examples

			For n = 1 there is only one free domino. For n = 2 there are 5 free tetrominoes. For n = 3 there are 35 free hexominoes. For n = 4 there are 369 free octominoes (see link section).

Links

Crossrefs

Bisection of A000105.
Cf. A000988, A056785, A056786, A210988, A210997, A216492, A216583.

Programs

Formula

a(n) = A000105(2n).
a(n) = A213376(n) + A056785(n). - R. J. Mathar, Feb 08 2023

Extensions

More terms from John Mason, Apr 15 2023

A213377 Number of polyominoes of order 2n that can be tiled by dominoes in a unique way.

Original entry on oeis.org

1, 3, 20, 170, 1728, 18878, 214278, 2488176, 29356463
Offset: 1

Views

Author

Joseph Myers, Jun 10 2012

Keywords

Comments

Tilings related by a symmetry of the polyomino that is not a symmetry of the tiling count as distinct (thus, the square tetromino counts as being tiled in two distinct ways).

Links

Crossrefs

Cf. A056785, A056786, A213376, A213378.

A213378 Number of polyominoes of order 2n that can be tiled by dominoes in more than one way.

Original entry on oeis.org

0, 1, 3, 41, 499, 6946, 95964, 1330807, 18395673
Offset: 1

Views

Author

Joseph Myers, Jun 10 2012

Keywords

Comments

Tilings related by a symmetry of the polyomino that is not a symmetry of the tiling count as distinct (thus, the square tetromino counts as being tiled in two distinct ways).

Crossrefs

Cf. A056785, A056786, A213376, A213377.

A252653 Number of free polyominoes with n cells containing a complete self-avoiding walk.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 46, 115, 300, 781, 2097, 5541
Offset: 1

Views

Author

Ernest French, Mar 21 2015

Keywords

Links

Crossrefs

Cf. A000105, A213376, A283162.

A342430 Number of prime polyominoes with n cells.

Original entry on oeis.org

0, 0, 1, 2, 1, 12, 5, 108, 145, 974, 2210, 17073, 31950, 238591, 587036, 3174686, 9236343, 50107909
Offset: 0

Views

Author

Drake Thomas, Mar 11 2021

Keywords

Comments

We say that a free polyomino is prime if it cannot be tiled by any other free polyomino besides the 1 X 1 square and itself.
The tiling of P must be with a single polyomino, and that single polyomino may not be the unique monomino or P itself. For example, decomposing the T-tetromino into a 3 X 1 and a 1 X 1 would use multiple tiles, and this is not permitted.
It can be shown that a(n) > 0 for all n >= 4, by considering the polyomino whose cells are at (0,1), (-1,1), (0,2), and (x,0) for all x = 0, 1, ..., n-4.

Examples

			For n = 4, the T-tetromino cannot be decomposed into smaller congruent polyominoes:
      +---+
      |   |
  +---+   +---+
  |           |
  +-----------+
The other four free tetrominoes can, however:
  +---+
  |   |
  |   |    +---+
  |   |    |   |
  +---+    |   |         +---+---+        +---+---+
  |   |    |   |         |   |   |        |       |
  |   |    +---+---+     |   |   |    +---+---+---+
  |   |    |       |     |   |   |    |       |
  +---+    +-------+     +---+---+    +---+---+
Thus a(4) = 1.

Links

Crossrefs

Cf. A000105, A125759, A213376.

Formula

a(n) = A000105(n) if n is prime.

Extensions

a(14)-a(17) from John Mason, Sep 16 2022
a(1) corrected by John Mason, Feb 27 2023
Showing 1-6 of 6 results.

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