A001168 -id:A001168 - cp's OEIS frontend

A338210 Triangle of coefficients of perimeter polynomials for fixed polyominoes.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 9, 8, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 28, 12, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 54, 80, 60, 16, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 136, 252, 228, 100, 20, 2

Offset: 0

Sean A. Irvine, Oct 16 2020

Considered as a triangle, T(n,k) is the number of polyominoes of n cells having a (cell) perimeter of k.

			Polynomials begin:
  1;
  x^4;
  2*x^6;
  4*x^7 + 2*x^8;
  9*x^8 + 8*x^9 + 2*x^10;
  ...

Sean A. Irvine, Table of n, a(n) for n = 0..437; rows 0..19 flattened
Sean A. Irvine, Java program (github)

Cf. A001168 (row sums), A338211 (free equivalent), A338212 (sprawl), A003203.

A001168(n) = Sum_{k=0..2*n+2} T(n,k).

A365651 Number of fixed n-polyominoids, allowing both corner- and edge-connections.

Original entry on oeis.org

3, 48, 1072, 27732, 781200

Offset: 1

Pontus von Brömssen, Sep 17 2023

Index entries for sequences related to polyominoes.

Cf. A001168 (polyominoes), A006770 (polyplets), A075678 (polyominoids), A365650 (free), A365653 (corner-connections only).

30th row of A366767.

A338212 a(n) is the number of fixed polyominoes with sprawl n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 4, 2, 9, 9, 22, 32, 66, 104, 200, 348, 646, 1172, 2144, 3964, 7285, 13581, 25136, 47070, 87700, 164764, 309308, 582152, 1098698, 2074484

Offset: 0

Sean A. Irvine and Allan C. Wechsler, Oct 17 2020

The sprawl of a polyomino is the number of cells in a polyomino plus the number of cells adjacent to it.

Sean A. Irvine, Java program (github)

Cf. A001168, A338210, A338213.

Antidiagonal sums of A338210.

A348848 Number of oriented polyominoes with 4n cells that have fourfold rotational symmetry centered at a vertex.

Original entry on oeis.org

1, 2, 6, 19, 65, 224, 790, 2851, 10424, 38496, 143454, 538667, 2035180, 7729146, 29486904, 112942373, 434114384, 1673766428, 6471199322, 25081542410, 97431694571, 379256586232, 1479022885116

Offset: 1

Robert A. Russell, Nov 01 2021

nonn

These are polyominoes of the regular tiling with Schläfli symbol {4,4}. For oriented polyominoes, chiral pairs are counted as two. This is one of the five sequences, along with A001168, needed to calculate the number of oriented polyominoes, A000988. It is the C90(n/4) sequence in the Shirakawa link. The calculation follows Redelmeier's method of inner rings.

			For a(1)=1, the polyomino is a 2 X 2 square. For a(2)=2, the two polyominoes are a chiral pair having a central 2 X 2 square with one cell attached to each edge of that square.

Robert A. Russell, Table of n, a(n) for n = 1..23
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Cf. A000988, A144553, A348849 (cell center).

Inner rings: A324406, A324407, A324408, A324409.

A348849 Number of fixed polyominoes with n cells that have fourfold rotational symmetry centered at the center of a cell.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 3, 6, 0, 0, 10, 18, 0, 0, 35, 57, 0, 0, 126, 191, 0, 0, 461, 658, 0, 0, 1699, 2308, 0, 0, 6315, 8241, 0, 0, 23686, 29853, 0, 0, 89432, 109268, 0, 0, 339473, 403450, 0, 0, 1294826, 1501074

Offset: 1

Robert A. Russell, Nov 01 2021

nonn

These are polyominoes of the regular tiling with Schläfli symbol {4,4}. Chiral pairs are counted as two. This is one of the five sequences, along with A001168, needed to calculate the number of oriented polyominoes, A000988. It is the F90 sequence in the Shirakawa link. The calculation follows Redelmeier's method of determining inner rings.

			For a(9)=2, the polyomino is a 3 X 3 square or a row and column of five cells sharing their central cells.

Robert A. Russell, Table of n, a(n) for n = 1..96
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Cf. A000988, A144553, A348848 (vertex center).

Inner rings: A324406, A324407, A324408, A324409.

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

A048664 Number of rooted 2-dimensional polyominoes with n square cells, with no symmetries removed.

Original entry on oeis.org

1, 4, 18, 76, 315, 1296, 5320, 21800, 89190, 364460, 1487948, 6070332, 24750570, 100868236, 410919990, 1673486992, 6813529348, 27734769756, 112874034844, 459295593200, 1868653768443, 7601716598916, 30920563717052, 125759730486432, 511445050400275, 2079809585544808, 8457048866652588, 34386482811155244

Offset: 1

Dan Hoey

nonn

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

Jean-François Alcover, Table of n, a(n) for n = 1..56
R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

Cf. A001168, A048663.

A row of A094166.

A001168 = Cases[Import["https://oeis.org/A001168/b001168.txt", "Table"], {, }][[All, 2]];
a[n_] := n A001168[[n]];
Array[a, 56] (* Jean-François Alcover, Sep 08 2019, after Andrew Howroyd *)

a(n) = n * A001168(n). - Andrew Howroyd, Dec 04 2018

More terms from Richard C. Schroeppel, May 04 2004

A056261 Series for radius of gyration for lattice animals on square lattice.

Original entry on oeis.org

2, 28, 252, 1840, 11924, 71476, 405204, 2202724, 11590162, 59417180, 298186524, 1470151308, 7140410208, 34237750548, 162350915772, 762391407024, 3549556185044, 16400558514664, 75263022053196

Offset: 2

N. J. A. Sloane, Aug 27 2000

nonn

I. Jensen, Table of n, a(n) for n = 2..42 (from link below)
I. Jensen, Enumerations of lattice animals and trees, J. Stat. Phys. 103 (3-4) (2001) 865-881, Table II.
I. Jensen, Enumerations of lattice animals and trees, arXiv:cond-mat/0007239.
I. Jensen, Terms a(2)..a(42)
I. Jensen, Home page

Cf. A001168.

A065068 Number of fixed polyominoes with n cells of which no four are equally spaced on a straight line.

Original entry on oeis.org

1, 2, 6, 17, 45, 114, 264, 545, 1061, 2070, 3990, 7275, 12556, 20824, 33584, 52753, 80560, 119274, 170168, 232700, 306472, 396580, 517300, 688736, 926816, 1241720, 1640968, 2130448, 2710356, 3383596, 4171788, 5120776, 6289760, 7747928, 9582196, 11908872, 14887792, 18732340, 23712492, 30141180, 38341172, 48622328, 61275752, 76578628, 94786504, 116132688, 140870564, 169379536, 202262720, 240326780, 284337040, 334532714, 389976092

Offset: 1

Jan Kristian Haugland, Nov 07 2001

nonn

a(142) = 36 is the final nonzero term of this sequence. - Jan Kristian Haugland, Apr 28 2020

Jan Kristian Haugland, Illustration for n = 142 [Cached copy, with permission]
J. K. Haugland, Largest polyomino with no four cells equally spaced on a straight line, arXiv:2004.12801 [math.GM] (2020).
Wikipedia, Polyomino

Cf. A001168.

a(n) = 0 for n >= 143. - Jan Kristian Haugland, Apr 28 2020

a(32)-a(53) from Brendan Owen (brendan_owen(AT)yahoo.com), Mar 11 2002

cp's OEIS Frontend

A338210 Triangle of coefficients of perimeter polynomials for fixed polyominoes.

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A365651 Number of fixed n-polyominoids, allowing both corner- and edge-connections.

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A338212 a(n) is the number of fixed polyominoes with sprawl n.

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A348848 Number of oriented polyominoes with 4n cells that have fourfold rotational symmetry centered at a vertex.

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A348849 Number of fixed polyominoes with n cells that have fourfold rotational symmetry centered at the center of a cell.

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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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Examples

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Formula

Extensions

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

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Programs

Mathematica

Formula

Extensions

A048664 Number of rooted 2-dimensional polyominoes with n square cells, with no symmetries removed.

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References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A056261 Series for radius of gyration for lattice animals on square lattice.

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Keywords

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Crossrefs

A065068 Number of fixed polyominoes with n cells of which no four are equally spaced on a straight line.

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Author