John Mason - cp's OEIS frontend

A385715 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional (n,2)-polyominoids, n >= 2, of size k >= 1.

1, 2, 3, 6, 18, 6, 19, 158, 60, 10, 63, 1611, 916, 140, 15, 216, 17811, 16698, 3060, 270, 21, 760, 207395, 336210, 81090, 7690, 462, 28, 2725, 2505858, 7218768, 2396434, 268005, 16226, 728, 36, 9910, 31125711, 162185112, 76020890, 10477161, 701589, 30408, 1080, 45

Offset: 2

John Mason, Jul 07 2025

			The top corner of the array (size on horizontal axis, dimensions on vertical):
              1    2     3       4         5          6           7           8         9         10
(A001168) 2:  1    2     6      19        63        216         760        2725      9910      36446
(A075678) 3:  3   18   158    1611     17811     207395     2505858    31125711 394982973 5098498323
(A366335) 4:  6   60   916   16698    336210    7218768   162185112  3769221330
          5: 10  140  3060   81090   2396434   76020890  2535403620 87781527395
          6: 15  270  7690  268005  10477161  441378400 19603138320
          7: 21  462 16226  701589  34160301 1796996509
          8: 28  728 30408 1570436  91583156
          9: 36 1080 52296 3141108 213477012

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Rows: A001168 (n=2), A075678 (n=3), A366335 (n=4).
Columns: A000217 (k=1), A213820 (k=2).
Cf. A385291 (polyominoes), A385581 (polysticks).

A385400 a(n) is the number of free polyominoids that have faces aligned to precisely 3 planes.

Original entry on oeis.org

0, 0, 2, 16, 239, 3154, 42225, 561178, 7459089

Offset: 1

John Mason, Jun 27 2025

John Mason, Illustration of size 4 polyominoids

Cf. A000105 (faces aligned to precisely 1 plane), A385399 (faces aligned to precisely 2 planes), A075679.

a(n) = A075679(n) - A385399(n) - A000105(n).

A385399 a(n) is the number of free polyominoids that have faces aligned to precisely 2 planes.

Original entry on oeis.org

0, 1, 5, 33, 197, 1461, 11278, 93486, 799261

Offset: 1

John Mason, Jun 27 2025

John Mason, Illustration of size 3 polyominoids

Cf. A000105 (faces aligned to precisely 1 plane), A385400 (faces aligned to precisely 3 planes), A075679.

a(n) = A075679(n) - A385400(n) - A000105(n).

A385291 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional polyominoes of size k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 19, 15, 4, 1, 1, 63, 86, 28, 5, 1, 1, 216, 534, 234, 45, 6, 1, 1, 760, 3481, 2162, 495, 66, 7, 1, 1, 2725, 23502, 21272, 6095, 901, 91, 8, 1, 1, 9910, 162913, 218740, 80617, 13881, 1484, 120, 9, 1, 1, 36446, 1152870, 2323730, 1121075, 231008, 27468, 2276, 153, 10, 1

Offset: 1

John Mason, Jun 24 2025

			The top corner of the array (size on horizontal axis, dimensions on vertical):
          1: 1  1    1     1       1         1           1
(A001168) 2: 1  2    6    19      63       216         760
(A001931) 3: 1  3   15    86     534      3481       23502
(A151830) 4: 1  4   28   234    2162     21272      218740
(A151831) 5: 1  5   45   495    6095     80617     1121075
(A151832) 6: 1  6   66   901   13881    231008     4057660
(A151833) 7: 1  7   91  1484   27468    551313    11710328
(A151834) 8: 1  8  120  2276   49204   1156688    28831384
(A151835) 9: 1  9  153  3309   81837   2205489    63113061
         10: 1 10  190  4615  128515   3906184   126210640
         11: 1 11  231  6226  192786   6524265   234919234
         12: 1 12  276  8174  278598  10389160   412504236
         13: 1 13  325 10491  390299  15901145   690185431
         14: 1 14  378 13209  532637  23538256  1108774772
         15: 1 15  435 16360  710760  33863201  1720467820
         16: 1 16  496 19976  930216  47530272  2590788848
         17: 1 17  561 24089 1196953  65292257  3800689609
         18: 1 18  630 28731 1517319  88007352  5448801768
         19: 1 19  703 33934 1898062 116646073  7653842998
         20: 1 20  780 39730 2346330 152298168 10557176740
         21: 1 21  861 46151 2869671 196179529 14325525627
         22: 1 22  946 53229 3476033 249639104 19153838572
         23: 1 23 1035 60996 4173764 314165809 25268311520
         24: 1 24 1128 69484 4971612 391395440 32929561864

John Mason, Table of n, a(n) for n = 1..162

Cf. A000384 (column k=3), A195739.

Rows: A000012 (n=1), A001168 (n=2), A001931 (n=3), A151830 (n=4), A151831 (n=5), A151832 (n=6), A151833 (n=7), A151834 (n=8), A151835 (n=9).

A(n,k) = Sum_{d=0..n} binomial(n,d)*A195739(k,d) (with A195739(k,d) = 0 for k <= d). - Pontus von Brömssen, Jun 28 2025

a(56)-a(66) from Pontus von Brömssen, Jun 28 2025

A381703 Irregular triangle read by rows in which every row of length A071764(n) lists A(n,w,h) = the number of free polyominoes of size n, width w and height h (for w <= h, and all possible w,h pairs).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 2, 3, 6, 1, 1, 6, 5, 7, 15, 1, 2, 11, 5, 7, 39, 25, 18, 1, 1, 10, 19, 7, 3, 59, 96, 35, 77, 61, 1, 3, 22, 28, 7, 1, 42, 210, 188, 49, 181, 383, 97, 73, 1, 1, 15, 52, 40, 9, 21, 255, 550, 332, 63, 266, 1304, 822, 155, 529, 240, 1, 3, 45, 90, 53, 9, 4, 212, 954, 1231, 529, 81, 251, 2847, 3548, 1551, 220, 2413, 2366, 410, 255

Offset: 1

John Mason, Mar 04 2025

			Triangle begins:
   n
   1:  1
   2:  1
   3:  1  1
   4:  1  1   3
   5:  1  2   3   6
   6:  1  1   6   5   7  15
   7:  1  2  11   5   7  39  25   18
   8:  1  1  10  19   7   3  59   96   35   77   61
   9:  1  3  22  28   7   1  42  210  188   49  181  383    97   73
  10:  1  1  15  52  40   9  21  255  550  332   63  266  1304  822  155  529  240
  ...
Any row contains an irregular array that shows the number of polyominoes having width w and height h. E.g., row 6 contains the array:
  h/w 1  2  3
  1
  2
  3      1  7
  4      6 15
  5      5
  6   1
.
There are 5 polyominoes of size 6 with width 2 and height 5, so A(6,2,5)=5:
.
  OO O  O  O  O
  O  OO O  O  O
  O  O  OO O  OO
  O  O  O  OO  O
  O  O  O   O  O

John Mason, Table of n, a(n) for n = 1..386

Row sums give A000105.
Row lengths give A071764.
Cf. A379623, A379624, A379625, A317186.

More terms from John Mason, Mar 07 2025

A381030 Array read by diagonals downwards: A(n,k) for n>=2 and k>=0 is the number of (n,k)-polyominoes.

Original entry on oeis.org

1, 2, 2, 2, 4, 5, 3, 11, 20, 12, 3, 17, 60, 68, 35, 4, 32, 151, 302, 289, 108, 4, 45, 322, 955, 1523, 1151, 369, 5, 71, 633, 2617, 5942, 7384, 4792, 1285, 5, 94, 1132, 6179, 19061, 33819, 35188, 19603, 4655, 6, 134, 1930, 13374, 52966, 125940, 184938, 164036, 80820, 17073, 6, 170, 3095, 26567, 131717, 400119, 778318, 969972

Offset: 2

John Mason, Feb 12 2025

(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, 2 different sets of the same number of transparent squares that connect in distinct ways the same set of visible squares, count as 1. E.g. these 2 different formations count as 1:
   XO  XOO
    OX   X

			The table begins as follows:
  n\k|     0      1      2      3      4      5      6     7    8   9 10
  ---+------------------------------------------------------------------
    2|     1      2      2      3      3      4      4     5    5   6  6
    3|     2      4     11     17     32     45     71    94  134 170
    4|     5     20     60    151    322    633   1132  1930 3095
    5|    12     68    302    955   2617   6179  13374 26567
    6|    35    289   1523   5942  19061  52966 131717
    7|   108   1151   7384  33819 125940 400119
    8|   369   4792  35188 184938 778318
    9|  1285  19603 164036 969972
   10|  4655  80820 753310
   11| 17073 331373
   12| 63600

Dmitry Kamenetsky and Tristrom Cooke, Tiling rectangles with holey polyominoes, arXiv:1411.2699 [cs.CG], 2015.

Cf. A381057.

Columns 0..4: A000105, A286344, A286194, A286345.

First row, a(2,k) = floor((k+3)/2).

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

John Mason, Feb 12 2025

(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

			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

Dmitry Kamenetsky and Tristrom Cooke, Tiling rectangles with holey polyominoes, arXiv:1411.2699 [cs.CG], 2015.

Row 2 gives A005418.
Column 0 gives A000105.
Cf. A381030.

A377593 Number of aligned fixed polyominoes that will fit in a square of size n X n.

Original entry on oeis.org

1, 8, 151, 9472, 2081051, 1643823600, 4742607132499, 50303895480064088, 1966122506151835674303, 283294196554063138439927568, 150432366492029200690537003170367, 294212995394376069103067524948055548348, 2117957146063247996594586658579155551318256103, 56084287855193446153928896349599388059636859288133588, 5460061052459125116800111315595463810654508452342242195388707

Offset: 1

John Mason, Nov 02 2024

nonn

a(n) is the number of fixed polyominoes that have both width and height <= n. The word "aligned" in the title refers to the restriction that the polyominoes have edges parallel to the sides of the square.

			a(2) = 8 because of the monomino, 2 alignments of the domino, 4 alignments of the L-shaped tromino, and the square tetromino.

Cf. A292357, A268416, A268404.

a(n) = Sum_{i=1..n,j=1..n} A292357(i,j).

A377155 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 1, 3, 4, 0, 8, 4, 3, 8, 3, 2, 17, 6, 8, 19, 27, 2, 53, 19, 26, 49, 19, 10, 127, 38, 64, 121, 166, 15, 373, 111, 197, 306, 150, 67, 923, 242, 460, 771, 1100, 115, 2665, 686, 1405, 1972, 1085, 431, 6681, 1562, 3335, 5051, 7353

Offset: 1

John Mason, Oct 18 2024

nonn

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice.

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Cf. A000162, A066273, A066281, A066283, A066287, A066288, A066453, A066454, A377128, A377129, A377130.

a(n) = 2*BD(n)+CCC(n)+DEE(n) = 2*A377128(n)+A377129(n)+A377130(n); see Lunnon paper for naming convention.

A377128 Number of polycubes of size n and symmetry class BD.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 1, 0, 0, 4, 2, 2, 3, 4, 0, 23, 7, 5, 10, 3, 1, 48, 16, 19, 28, 49, 2, 174, 49, 58, 84, 46, 18, 406, 111, 169, 238, 424, 34, 1285, 321, 524, 678, 410, 153, 3139, 747, 1393, 1872, 3185

Offset: 1

John Mason, Oct 17 2024

nonn

See link "Counting free polycubes" for explanation of notation.

John Mason, Counting free polycubes

Cf. A000162, A038119.

cp's OEIS Frontend

User: John Mason

A385715 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional (n,2)-polyominoids, n >= 2, of size k >= 1.

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A385400 a(n) is the number of free polyominoids that have faces aligned to precisely 3 planes.

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A385399 a(n) is the number of free polyominoids that have faces aligned to precisely 2 planes.

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A385291 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional polyominoes of size k.

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A381703 Irregular triangle read by rows in which every row of length A071764(n) lists A(n,w,h) = the number of free polyominoes of size n, width w and height h (for w <= h, and all possible w,h pairs).

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A381030 Array read by diagonals downwards: A(n,k) for n>=2 and k>=0 is the number of (n,k)-polyominoes.

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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.

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A377593 Number of aligned fixed polyominoes that will fit in a square of size n X n.

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A377155 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 12.

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A377128 Number of polycubes of size n and symmetry class BD.

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