A151831 -id:A151831 - cp's OEIS frontend

A151830 Number of fixed 4-dimensional polycubes with n cells.

1, 4, 28, 234, 2162, 21272, 218740, 2323730, 25314097, 281345096, 3178474308, 36400646766, 421693622520, 4933625049464, 58216226287844, 692095652493483

Offset: 1

N. J. A. Sloane, Jul 12 2009

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275.
S. Luther and S. Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), 546-565.

G. Aleksandrowicz and G. Barequet, counting d-dimensional polycubes and nonrectangular planar polyomnoes, Lect. Not. Comp. Sci 4112 (2006) 418-427 Table 2
Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).

Cf. A001931, A068870 (free), A151831, A151832, A151833, A151834, A151835.

a(16) from Luther and Mertens by Gill Barequet, Jun 12 2011

A151832 Number of fixed 6-dimensional polycubes with n cells.

Original entry on oeis.org

1, 6, 66, 901, 13881, 231008, 4057660, 74174927, 1398295989, 27012396022, 532327974882, 10665521789203, 216696065279573, 4455636282185802, 92567760074841818

Offset: 1

N. J. A. Sloane, Jul 12 2009

Anthony J. Guttmann, editor. Polygons, Polyominoes and Polycubes, volume 775 of Lecture Notes in Physics. Springer-Verlag, Heidelberg, 2009.

J. Adler, Y. Meir, A. B. Harris, A. Aharony, and J. A. M. S. Duarté, Series study of random animals in general dimensions, Physical Review B, 38 (1988) 4941.
Gadi Aleksandrowicz and Gill Barequet, Counting polycubes without the dimensionality curse, Discrete Mathematics, 309 (2009), 4576-4583.
Gadi Aleksandrowicz and Gill Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
Gadi Aleksandrowicz and Gill Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.
Gill Barequet, Gil Ben-Shachar, and Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry (Würzburg, Germany, 16-18 March 2020).
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016.
Ronnie Barequet, Gill Barequet, and Günter Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30, 257-275 (2010).
D. S. Gaunt and P. J. Peard, 1/d-expansions for the free energy of weakly embedded site animal models of branched polymers, Journal of Physics A: Mathematical and General, 33 (2000) 7515-7539.
Hsiao-Ping Hsu, Walter Nadler, and Peter Grassberger, Statistics of lattice animals, Computer Physics Communications, 169 (2005) 114-116.
Iwan Jensen, Enumerations of lattice animals and trees, Journal of Statistical Physics, 102(3/4) (2001) 865-881.
Sebastian Luther and Stephan Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), 546-565. See Table 2 for the terms (but beware of the incorrect a(13)!) or Table 3 for the formulas.
Sebastian Luther and Stephan Mertens, Counting lattice animals in high dimensions, arXiv:1106.1078 [cond-mat.stat-mech], 2011.
Stephan Mertens and Markus E. Lautenbacher, Counting lattice animals: A parallel attack, J. Stat. Phys. 66 (1992) 669.

Cf. A001931, A048667, A151830, A151831, A151833, A151834, A151835.

A048667 = Cases[Import["https://oeis.org/A048667/b048667.txt", "Table"], {, }][[All, 2]];
a[n_] := A048667[[n]]/n;
Array[a, 15] (* Jean-François Alcover, Sep 12 2019 *)

a(n) = A048667(n)/n. - Jean-François Alcover, Sep 12 2019, after Andrew Howroyd in A048667.

a(10) from Gadi Aleksandrowicz (gadial(AT)gmail.com), Mar 21 2010
a(11)-a(15) from Luther and Mertens by Gill Barequet, Jun 12 2011
a(13) corrected by M. F. Hasler, Jun 26 2025

A151833 Number of fixed 7-dimensional polycubes with n cells.

Original entry on oeis.org

1, 7, 91, 1484, 27468, 551313, 11710328, 259379101, 5933702467, 139272913892, 3338026689018, 81406063278113, 2014611366114053, 50486299825273271

Offset: 1

N. J. A. Sloane, Jul 12 2009

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
G. Aleksandrowicz and G. Barequet, Counting polycubes without the dimensionality curse, Discrete Mathematics, 309 (2009), 4576-4583.
G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275.
S. Luther and S. Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), 546-565.

Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).

Cf. A001931, A048668, A151830, A151831, A151832, A151834, A151835.

A048668 = Cases[Import["https://oeis.org/A048668/b048668.txt", "Table"], {, }][[All, 2]];
a[n_] := A048668[[n]]/n;
Array[a, 14] (* Jean-François Alcover, Sep 12 2019 *)

a(n) = A048668(n)/n. - Jean-François Alcover, Sep 12 2019, after Andrew Howroyd in A048668.

More terms from Gadi Aleksandrowicz (gadial(AT)gmail.com), Mar 21 2010

a(11)-a(14) from Luther and Mertens by Gill Barequet, Jun 12 2011

A151834 Number of fixed 8-dimensional polycubes with n cells.

Original entry on oeis.org

1, 8, 120, 2276, 49204, 1156688, 28831384, 750455268, 20196669078, 558157620384, 15762232227968, 453181069339660, 13228272325440164, 391166062869849024

Offset: 1

N. J. A. Sloane, Jul 12 2009

a(1)-a(10) can be computed by formulas in Barequet et al. (2010). Luther and Mertens confirm these values (and add two more) by direct counting.

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
G. Aleksandrowicz and G. Barequet, Counting polycubes without the dimensionality curse, Discrete Mathematics, 309 (2009), 4576-4583.
G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.
Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes; in: Handbook of Discrete and Computational Geometry, Chapman and Hall/CRC, 2017. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016.
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275.
S. Luther and S. Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), P09026.
Stephan Mertens, Lattice Animals

Cf. A001931, A151830, A151831, A151832, A151833, A151835.

More terms from Gadi Aleksandrowicz (gadial(AT)gmail.com), Mar 21 2010
a(9)-a(12) from Luther and Mertens by Gill Barequet, Jun 12 2011
a(13)-a(14) from Mertens added by Andrey Zabolotskiy, Jan 29 2023

A151835 Number of fixed 9-dimensional polycubes with n cells.

Original entry on oeis.org

1, 9, 153, 3309, 81837, 2205489, 63113061, 1887993993, 58441956579, 1858846428437, 60445700665383, 2001985304489169, 67341781440810531, 2295424989986481345

Offset: 1

N. J. A. Sloane, Jul 12 2009

a(1)-a(10) can be computed by formulas in Barequet et al. (2010). Luther and Mertens confirm these values (and add two more) by direct counting.

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
Gill Barequet, Gil Ben-Shachar, and Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes; in: Handbook of Discrete and Computational Geometry, Chapman and Hall/CRC, 2017. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016.
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275.
S. Luther and S. Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), P09026.
Stephan Mertens, Lattice Animals

Cf. A001931, A151830, A151831, A151832, A151833, A151834.

a(5)-a(12) from Luther and Mertens by Gill Barequet, Jun 12 2011

a(13)-a(14) from Mertens added by Andrey Zabolotskiy, Jan 29 2023

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

A048666 Number of rooted 5-dimensional "polycubes" with n cells, with no symmetries removed.

Original entry on oeis.org

1, 10, 135, 1980, 30475, 483702, 7847525, 129419240, 2161766520, 36481155310, 620845213890, 10640356142700, 183453873965570, 3179310190998270, 55345614317169210

Offset: 1

Dan Hoey

nonn

R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

Row 5 of A048790.

Cf. A048663, A151831.

a(n) = n * A151831(n). - Andrew Howroyd, Dec 05 2018

a(9)-a(15) from Andrew Howroyd, Dec 05 2018

cp's OEIS Frontend

A151830 Number of fixed 4-dimensional polycubes with n cells.

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A151832 Number of fixed 6-dimensional polycubes with n cells.

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A151833 Number of fixed 7-dimensional polycubes with n cells.

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A151834 Number of fixed 8-dimensional polycubes with n cells.

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A151835 Number of fixed 9-dimensional polycubes with n cells.

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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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A048666 Number of rooted 5-dimensional "polycubes" with n cells, with no symmetries removed.

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