A195739 -id:A195739 - cp's OEIS frontend

A259015 The number of fixed polycubes of size n that span n-4 dimensions.

0, 1, 214, 21225, 1688424, 125055400, 9178531200, 687848686448, 53435249786880, 4336107249936384, 368887991492608000, 32948013484980000000, 3090086319932923969536, 304136142049322287011840, 31382704663810285705887744, 3390841628447041935421747200, 383124440688361472000000000000

Offset: 4

Mira Shalah, Jun 16 2015

G. Barequet, M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG’15). Editors: Lars Arge and János Pach; pp. 19-22, 2015.

Charles R Greathouse IV, Table of n, a(n) for n = 4..351
G. Barequet and M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes (video)
G. Barequet and M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes (pdf file)

Diagonal 4 of A195739.

[2^(n-7)*n^(n-9)*(n-4)*(8*n^8-128*n^7+828*n^6-2930*n^5 +7404*n^4-17523*n^3+41527*n^2-114302*n+204960)/6: n in [4..20]]; // Vincenzo Librandi, Jun 20 2015

Table[2^(n - 7) n^(n - 9) (n - 4) (8 n^8 - 128 n^7 + 828 n^6 - 2930 n^5 + 7404 n^4 - 17523 n^3 + 41527 n^2 - 114302 n + 204960)/6, {n, 4, 20}] (* Michael De Vlieger, Jun 19 2015 *)

a(n)=2^(n-7)*n^(n-9)*(n-4)*(8*n^8-128*n^7+828*n^6 -2930*n^5 +7404*n^4-17523*n^3 +41527*n^2-114302*n +204960)/6 \\ Charles R Greathouse IV, Jun 16 2015

a(n) = 2^(n-7)*n^(n-9)*(n-4)*(8*n^8 - 128*n^7 + 828*n^6 - 2930*n^5 + 7404*n^4 - 17523*n^3 + 41527*n^2 - 114302*n + 204960)/6.

Typo in formula fixed by Colin Barker, Jun 16 2015

A385582 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional polysticks of size n.

Original entry on oeis.org

1, 1, 4, 1, 20, 32, 1, 86, 420, 400, 1, 370, 4164, 10368, 6912, 1, 1626, 38205, 186552, 301840, 153664, 1, 7310, 343380, 2934560, 8637760, 10223616, 4194304, 1, 33464, 3086049, 43517697, 207353960, 427708848, 396809280, 136048896

Offset: 1

Pontus von Brömssen, Jul 04 2025

			Triangle begins:
  n\d| 1     2       3        4         5         6         7         8
  ---+-----------------------------------------------------------------
  1  | 1
  2  | 1     4
  3  | 1    20      32
  4  | 1    86     420      400
  5  | 1   370    4164    10368      6912
  6  | 1  1626   38205   186552    301840    153664
  7  | 1  7310  343380  2934560   8637760  10223616   4194304
  8  | 1 33464 3086049 43517697 207353960 427708848 396809280 136048896

Pontus von Brömssen, Table of n, a(n) for n = 1..78 (first 12 rows)
Stephan Mertens and Cristopher Moore, Series expansion of the percolation threshold on hypercubic lattices, J. Phys. A: Math. Theor., 51 (2018), 475001; arXiv:1805.02701 [cond-mat.stat-mech], 2018.
Index entries for sequences related to polyominoes.

Cf. A127670 (main diagonal), A195739 (polyominoes), A365566 (free), A385581.

T(n,d) = Sum_{k=1..d} (-1)^(d-k)*binomial(d,k)*A385581(n,k).

A355997 Triangle read by rows: T(n,d) is the number of fixed orthoplex n-ominoes with cell centers determining d-space.

Original entry on oeis.org

1, 0, 4, 0, 1, 32, 0, 0, 48, 400, 0, 0, 28, 1728, 6912, 0, 0, 8, 4240, 62720, 153664, 0, 0, 1, 7272, 344320, 2457600, 4194304, 0, 0, 0, 8720, 1465600, 23872320, 105815808, 136048896, 0, 0, 0, 7136, 5254576, 182691200, 1603840000, 5017600000, 512000000

Offset: 2

Robert A. Russell, Jul 22 2022

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. Two fixed polyominoes are identical only if one is a translation of the other.

Conjecture: T(n,n-4) = 2^(n-7) * n^(n-9) * (n-4) * (n-5) * (n-6) * (n^6-14*n^5+65*n^4-189*n^3+594*n^2-1305*n+6832) / 6 ~ A259015(n) / 8.

			Triangle begins with T(2,1):
n\d 1 2  3    4       5         6          7          8         9
2   1
3   0 4
4   0 1 32
5   0 0 48  400
6   0 0 28 1728    6912
7   0 0  8 4240   62720    153664
8   0 0  1 7272  344320   2457600    4194304
9   0 0  0 8720 1465600  23872320  105815808  136048896
10  0 0  0 7136 5254576 182691200 1603840000 5017600000 512000000

Robert A. Russell, Table of n, a(n) for n = 2..73

Cf. A195739 (multidimensional).

Diagonals (with formulas) are A127670, A355998, A355999.

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

A006763 Number of fixed properly-3-dimensional polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 32, 348, 2836, 21225, 154741, 1123143, 8185403, 60088748, 444688325, 3317057654, 24925158492, 188543716451, 1434760675947, 10976610064899, 84379534826376, 651441493579872

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. F. Lunnon, Counting multidimensional polyominoes. Computer Journal 18 (1975), no. 4, pp. 366-367.

A column of A195739.

A001931 = Cases[Import["https://oeis.org/A001931/b001931.txt", "Table"], {, }][[All, 2]];
A001168 = Cases[Import["https://oeis.org/A001168/b001168.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 1, 0, A001931[[n]] - 3 (A001168[[n]] - 1)];
a /@ Range[1, 19] (* Jean-François Alcover, Sep 22 2019 *)

a(n) = A001931(n) - 3 * (A001168(n) - 1) for n > 1. - Sean A. Irvine, Jul 27 2017

More terms from Jean-François Alcover, Sep 22 2019

A387002 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

Original entry on oeis.org

1, 2, 12, 6, 140, 320, 19, 1554, 10368, 13520, 63, 17622, 265344, 892864, 786432, 216, 206747, 6390484, 41998840, 89389920, 58383808, 760, 2503578, 152166240, 1749529040, 6773387520

Offset: 1

Pontus von Brömssen, Aug 14 2025

A (d,2)-polyominoid consists of unit square cells with integer coordinates in the d-dimensional grid, where two cells are connected if they share an edge. The polyominoid is properly d-dimensional if it is not contained in a (d-1)-dimensional affine subspace.

			Triangle begins:
  n\d |     2          3           4           5          6        7  8  9 10 11
  ----+-------------------------------------------------------------------------
   1  |     1
   2  |     2         12
   3  |     6        140         320
   4  |    19       1554       10368       13520
   5  |    63      17622      265344      892864     786432
   6  |   216     206747     6390484    41998840   89389920 58383808
   7  |   760    2503578   152166240  1749529040 6773387520        ?  ?
   8  |  2725   31117536  3644734836 69246650605          ?        ?  ?  ?
   9  |  9910  394953243 88344741448           ?          ?        ?  ?  ?  ?
  10  | 36446 5098388985           ?           ?          ?        ?  ?  ?  ?  ?

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

Cf. A001168 (column d=2), A195739 (polyominoes), A385582 (polysticks), A385715, A387004 (free).

T(n,d) = Sum_{k=2..d} (-1)^(d-k)*binomial(d,k)*A385715(k,n), i.e., the n-th row is the inverse binomial transform of the n-th column of A385715 (with the convention that T(n,d) = A385715(d,n) = 0 when d <= 1).

A006764 Number of fixed properly-4-dimensional polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 0, 400, 8640, 129288, 1688424, 20762073, 248384816, 2937307716, 34618858308

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-67.

Cf. A000105. A column of A195739.

a(n) = A151830(n) - 4 * A006763(n) - 6 * A006762(n) - 4 for n > 1. - Sean A. Irvine, Jul 28 2017

cp's OEIS Frontend

A259015 The number of fixed polycubes of size n that span n-4 dimensions.

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A385582 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional polysticks of size n.

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A355997 Triangle read by rows: T(n,d) is the number of fixed orthoplex n-ominoes with cell centers determining d-space.

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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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A006763 Number of fixed properly-3-dimensional polyominoes with n cells.

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A387002 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

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A006764 Number of fixed properly-4-dimensional polyominoes with n cells.

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