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 1-4 of 4 results.

A286194 Number of (n,2)-polyominoes.

Original entry on oeis.org

0, 2, 11, 60, 302, 1523, 7384, 35188, 164036, 753310
Offset: 1

Views

Author

Dmitry Kamenetsky, May 05 2017

Keywords

Comments

(n,k)-polyominoes are disconnected polyominoes with n visible squares and k transparent squares. Importantly, k must be the least number of transparent squares that need to be converted to visible squares to make all the visible squares connected. Note that a regular polyomino of order n is a (n,0)-polyomino, since all its visible squares are already connected. For more details see the paper by Kamenetsky and Cooke.

Examples

			We can represent these polyominoes as binary matrices, where 1 means visible square and 0 means transparent square. Note that we need to flip (change to 1) two 0's to make all the 1's connected. This also means that the Manhattan distance between any pair of 1's is at most 3. Here are all such polyominoes for n=2:
   1001 100
        001

Links

Crossrefs

Cf. A286344, A286345, A381030.

Extensions

a(6) corrected and a(7)-a(10) from John Mason, Feb 15 2025

A286344 Number of (n,1)-polyominoes.

Original entry on oeis.org

0, 2, 4, 20, 68, 289, 1151, 4792, 19603, 80820, 331373
Offset: 1

Views

Author

Dmitry Kamenetsky, May 07 2017

Keywords

Comments

(n,k)-polyominoes are disconnected polyominoes with n visible squares and k transparent squares. Importantly, k must be the least number of transparent squares that need to be converted to visible squares to make all the visible squares connected. Note that a regular polyomino of order n is a (n,0)-polyomino, since all its visible squares are already connected. For more details see the paper by Kamenetsky and Cooke.
Number of distinct n-cell subsets of (n+1)-celled polyominoes that are not polyominoes. - Charlie Neder, Feb 12 2019

Examples

			We can represent these polyominoes as binary matrices, where 1 means visible square and 0 means transparent square. Note that we need to flip (change to 1) one 0 to make all the 1s connected. This also means that the Manhattan distance between any pair of 1s is at most 2. Here are all such polyominoes for n=3:
   1101 100 100 010
        101 011 101

Links

Crossrefs

Cf. A286194, A286345, A381030.

Extensions

a(6)-a(7) corrected and a(8)-a(11) added by John Mason, Feb 15 2025

A286345 Number of (n,3)-polyominoes.

Original entry on oeis.org

0, 3, 17, 151, 955, 5942, 33819, 184938, 969972
Offset: 1

Views

Author

Dmitry Kamenetsky, May 07 2017

Keywords

Comments

(n,k)-polyominoes are disconnected polyominoes with n visible squares and k transparent squares. Importantly, k must be the least number of transparent squares that need to be converted to visible squares to make all the visible squares connected. Note that a regular polyomino of order n is a (n,0)-polyomino, since all its visible squares are already connected. For more details see the paper by Kamenetsky and Cooke.

Examples

			We can represent these polyominoes as binary matrices, where 1 means visible square and 0 means transparent square. Note that we need to flip (change to 1) three 0's to make all the 1's connected. This also means that the Manhattan distance between any pair of 1's is at most 4. Here are all such polyominoes for n=2:
   10001 1000 100
         0001 000
              001

Links

Crossrefs

Cf. A286194, A286344, A381030.

Extensions

a(6)-a(9) from John Mason, Feb 15 2025

A381057 Array read by diagonals downwards: A(n,k) for n>=2 and k>=0 is the number of (n,k)-polyominoes counting as distinct different formations of transparent squares.

Original entry on oeis.org

1, 2, 2, 3, 5, 5, 6, 17, 24, 12, 10, 41, 101, 89, 35, 20, 106, 353, 535, 382, 108, 36, 243, 1091, 2355, 2769, 1566, 369, 72, 567, 3095, 8937, 14841, 13739, 6569, 1285, 136, 1259, 8209, 29744, 65651, 86322, 66499, 27205, 4655, 272, 2806, 20804, 90914, 252277, 439879, 479343, 314445, 112886, 17073, 528, 6113, 50801, 259078, 872526
Offset: 2

Views

Author

John Mason, Feb 12 2025

Keywords

Comments

(n,k)-polyominoes are disconnected polyominoes with n visible squares and k transparent squares. Importantly, k must be the least number of transparent squares that need to be converted to visible squares to make all the visible squares connected. Note that a regular polyomino of order n is a (n,0)-polyomino, since all its visible squares are already connected. For more details see the paper by Kamenetsky and Cooke.
Note that, in this sequence, different sets of the same number of transparent squares that connect in distinct ways the same set of visible squares, are counted separately. E.g. these 2 different formations count as 2:
XO XOO
OX X

Examples

			The table begins as follows:
  n\k|     0      1       2       3       4       5      6      7     8    9  10
   --+--------------------------------------------------------------------------
    2|     1      2       3       6      10      20     36     72   136  272 528
    3|     2      5      17      41     106     243    567   1259  2806 6113
    4|     5     24     101     353    1091    3095   8209  20804 50801
    5|    12     89     535    2355    8937   29744  90914 259078
    6|    35    382    2769   14841   65651  252277 872526
    7|   108   1566   13739   86322  439879 1917387
    8|   369   6569   66499  479343 2759969
    9|  1285  27205  314445 2555903
   10|  4655 112886 1461335
   11| 17073 466178
   12| 63600

Links

Crossrefs

Row 2 gives A005418.
Column 0 gives A000105.
Cf. A381030.
Showing 1-4 of 4 results.

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