A036364 -id:A036364 - cp's OEIS frontend

A049430 Triangle read by rows: T(n,d) is the number of distinct properly d-dimensional polyominoes (or polycubes) with n cells (n >= 1, d >= 0).

1, 0, 1, 0, 1, 1, 0, 1, 4, 2, 0, 1, 11, 11, 3, 0, 1, 34, 77, 35, 6, 0, 1, 107, 499, 412, 104, 11, 0, 1, 368, 3442, 4888, 2009, 319, 23, 0, 1, 1284, 24128, 57122, 36585, 8869, 951, 47, 0, 1, 4654, 173428, 667959, 647680, 231574, 36988, 2862, 106, 0, 1, 17072, 1262464, 7799183, 11173880, 5712765, 1297366, 146578, 8516, 235

Offset: 1

Richard C. Schroeppel

			Triangle begins:
1
0 1
0 1     1
0 1     4       2
0 1    11      11       3
0 1    34      77      35        6
0 1   107     499     412      104      11
0 1   368    3442    4888     2009     319      23
0 1  1284   24128   57122    36585    8869     951     47
0 1  4654  173428  667959   647680  231574   36988   2862  106
0 1 17072 1262464 7799183 11173880 5712765 1297366 146578 8516 235
...

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

Cf. A049429 (col. d=0 omitted), A195738 (oriented), A195739 (fixed).
Row sums give A005519. Columns give A006765, A006766, A006767, A006768.
Diagonals (with algorithms) are A000055, A036364, A355053.
Cf. A330891 (cumulative sums of the rows).

Edited by N. J. A. Sloane, Sep 23 2011

More terms from John Niss Hansen, Mar 31 2015

A195738 Triangle read by rows: DR(n,d) is the number of properly d-dimensional polyominoes with n cells, modulo translations and rotations (n >= 1, 0 <= d <= n-1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 6, 3, 0, 1, 17, 17, 4, 0, 1, 59, 131, 52, 7, 0, 1, 195, 915, 709, 153, 13, 0, 1, 703, 6553, 8946, 3350, 454, 28

Offset: 1

N. J. A. Sloane, Sep 22 2011

From Petros Hadjicostas, Jan 11 2019: (Start)
Table 1 (p. 366) in Lunnon (1975) contains more terms. Because the table there (in the reference) has incomplete columns, the extra terms do not appear in this triangular sequence (array).
Entry DR(n=11, d=2) in Table 1 (p. 366) must be a typo. It should not be 33890, but 33895. This was corrected by N. J. A. Sloane in 2011 in the documentation of sequence A006758. (See also sequence A000988.)
(End)
The number of oriented polyominoes (chiral pairs counted as two) here is the sum of the number of unoriented polyominoes (chiral pairs counted as one) in A049430 and the number of chiral pairs. - Robert A. Russell, May 03 2020

			Triangle begins:
n\d| 0    1    2    3    4    5    6    7
---+---------------------------------=---
1  | 1
2  | 0    1
3  | 0    1    1
4  | 0    1    6    3
5  | 0    1   17   17    4
6  | 0    1   59  131   52    7
7  | 0    1  195  915  709  153   13
8  | 0    1  703 6553 8946 3350  454   28
...

W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (4) (1975) 366-367.

Columns give A006758 (and A000988), A006759, A006760, A006761.
Cf. A195739, A049430.
Cf. A000055, A045649, A036364, A036365.

From Robert A. Russell, May 03 2020: (Start)
For n > 1, DR(n,n-1) = A000055(n) + A045649(n).
DR(n,n-2) = A036364(n) + A036365(n).
We can add unoriented and chiral pairs for the top two diagonals. The summands have quick algorithms. (End)

Sequence corrected by Petros Hadjicostas, Jan 11 2019 after observation by Jon E. Schoenfield

A049429 Triangle T(n,d) = number of distinct d-dimensional polyominoes (or polycubes) with n cells (0 < d < n).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 11, 11, 3, 1, 34, 77, 35, 6, 1, 107, 499, 412, 104, 11, 1, 368, 3442, 4888, 2009, 319, 23, 1, 1284, 24128, 57122, 36585, 8869, 951, 47, 1, 4654, 173428, 667959, 647680, 231574, 36988, 2862, 106, 1, 17072, 1262464, 7799183, 11173880, 5712765, 1297366, 146578, 8516, 235

Offset: 2

Richard C. Schroeppel

These are unoriented polyominoes of the regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). For unoriented polyominoes, chiral pairs are counted as one. The dimension of the convex hull of the cell centers determines the dimension d. - Robert A. Russell, Aug 09 2022

			From _Robert A. Russell_, Aug 09 2022: (Start)
Triangle begins with T(2,1):
n\d 1     2       3       4        5       6       7      8    9  10
2   1
3   1     1
4   1     4       2
5   1    11      11       3
6   1    34      77      35        6
7   1   107     499     412      104      11
8   1   368    3442    4888     2009     319      23
9   1  1284   24128   57122    36585    8869     951     47
10  1  4654  173428  667959   647680  231574   36988   2862  106
11  1 17072 1262464 7799183 11173880 5712765 1297366 146578 8516 235
(End)

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

R. C. Schroeppel, A few mathematical experiments

Cf. A049430 (col. d=0 added), A195738 (oriented), A195739 (fixed).
Diagonals (with algorithms) are A000055, A036364, A355053.
Row sums give A005519. Columns are A006765-A006768.

Two more rows added by Robert A. Russell, Aug 09 2022.

A355053 Number of unoriented multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

1, 11, 77, 412, 2009, 8869, 36988, 146578, 560498, 2078927, 7530385, 26734692, 93360884, 321454484, 1093599885, 3681897625, 12284317088, 40660245162, 133638662066, 436488290069, 1417680926923, 4581355626106

Offset: 4

Robert A. Russell, Jun 16 2022

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). For unoriented polyominoes, chiral pairs are counted as one.

			a(4)=1 as there is only one tetromino in one-space. a(5)=11 because there are 5 achiral and 6 chiral pairs of pentominoes in 2-space, excluding the 1-D straight pentomino.

Robert A. Russell, Table of n, a(n) for n = 4..100
W. F. Lunnon, Counting multidimensional polyominoes. Computer Journal 18 (1975), no. 4, pp. 366-367.
Robert A. Russell, Trunk Generating Functions

Cf. A355052 (oriented), A355054 (chiral), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A000081 (rooted trees), A049430 (Lunnon's DE).

Other dimensions: A036364 (n-2), A000055 (n-1), A355048 (orthoplex).

sb[n_,k_]:= sb[n,k] = b[n+1-k,1] + If[n<2k, 0, sb[n-k,k]];
b[1,1] := 1; b[n_,1] := b[n,1] = Sum[b[i,1]sb[n-1,i]i,{i,1,n-1}]/(n-1);
b[n_,k_] := b[n,k] = Sum[b[i,1]b[n-i,k-1],{i,1,n-1}];
nmax = 30; B[x_] := Sum[b[i,1]x^i,{i,0,nmax}]
Drop[CoefficientList[Series[(50B[x]^6+3B[x]^7+30B[x]^2B[x^2]^2+3B[x]^3B[x^2](6+B[x^2])+3B[x]^5(37+2B[x^2])+12B[x]^4(1+3B[x^2])+B[x](57B[x^2]^2+6B[x^2]^3+6B[x^4]+6B[x^2]B[x^4])+4(3B[x^2]^2+11B[x^2]^3+B[x^3]^2+B[x^6]))/24+B[x]^2(112B[x]^5+9B[x]^6+3B[x^2]^2+4B[x]B[x^2]^2+B[x]^2B[x^2](14+B[x^2])+8B[x]^3(1+4B[x^2])+B[x]^4(167+10B[x^2]))/(8(1-B[x]))+B[x]^5(46B[x]^3+6B[x]^4+3B[x^2]+B[x]^2(67+2B[x^2])+B[x](2+6B[x^2]))/(2(1-B[x])^2)+B[x]^6(153B[x]^2+75B[x]^3+12B[x]^4+3B[x^2]+B[x](4+3B[x^2]))/(6(1-B[x])^3)+B[x]^9(21+4B[x])/(2(1-B[x])^4)+3B[x]^10/(2(1-B[x])^5)+B[x^2](6B[x]^3B[x^2]+2B[x]^4B[x^2]+13B[x^2]^2+19B[x^2]^3+2B[x]^2B[x^2](1+3B[x^2])+B[x^4]+B[x^2]B[x^4]+B[x](35B[x^2]^2+5B[x^2]^3+B[x^4]+B[x^2]B[x^4]))/(4(1-B[x^2]))+B[x^2]^4(5+3B[x^2]+B[x](8+B[x^2]))/(1-B[x^2])^2+2B[x^2]^5(1+B[x])/(1-B[x^2])^3+2B[x]B[x^3]^2/(6(1-B[x^3]))+B[x]B[x^4]^2/(2(1-B[x^4]))+B[x]^2B[x^2]^2(7B[x]^2+5B[x]^3+3B[x^2]+B[x](2+B[x^2]))/(2(1-B[x])(1-B[x^2]))+B[x]^5B[x^2]^2(3+2B[x])/((1-B[x])^2(1-B[x^2]))+B[x]^6B[x^2]^2/((1-B[x])^3(1-B[x^2]))+B[x]^2B[x^2]^4/((1-B[x])(1-B[x^2])^2)+B[x^2]B[x^4]^2(1+B[x])/(2(1-B[x^2])(1-B[x^4])),{x,0,nmax}],x],4]

a(n) = A355052(n) - A355054(n) = (A355052(n) + A355055(n)) / 2 = A355054(n) + A355055(n).
a(n) = A049430(n,n-3), the third diagonal of Lunnon's DE array.
G.f.: (50B(x)^6+3B(x)^7+30B(x)^2B(x^2)^2+3B(x)^3B(x^2)(6+B(x^2))+3B(x)^5(37+2B(x^2))+12B(x)^4(1+3B(x^2))+B(x)(57B(x^2)^2+6B(x^2)^3+6B(x^4)+6B(x^2)B(x^4))+4(3B(x^2)^2+11B(x^2)^3+B(x^3)^2+B(x^6)))/24 + B(x)^2(112B(x)^5+9B(x)^6+3B(x^2)^2+4B(x)B(x^2)^2+B(x)^2B(x^2)(14+B(x^2))+8B(x)^3(1+4B(x^2))+B(x)^4(167+10B(x^2)))/(8(1-B(x))) + B(x)^5(46B(x)^3+6B(x)^4+3B(x^2)+B(x)^2(67+2B(x^2))+B(x)(2+6B(x^2)))/(2(1-B(x))^2) + B(x)^6(153B(x)^2+75B(x)^3+12B(x)^4+3B(x^2)+B(x)(4+3B(x^2)))/(6(1-B(x))^3) + B(x)^9(21+4B(x))/(2(1-B(x))^4) + 3B(x)^10/(2(1-B(x))^5) + B(x^2)(6B(x)^3B(x^2)+2B(x)^4B(x^2)+13B(x^2)^2+19B(x^2)^3+2B(x)^2B(x^2)(1+3B(x^2))+B(x^4)+B(x^2)B(x^4)+B(x)(35B(x^2)^2+5B(x^2)^3+B(x^4)+B(x^2)B(x^4)))/(4(1-B(x^2))) + B(x^2)^4(5+3B(x^2)+B(x)(8+B(x^2)))/(1-B(x^2))^2 + 2B(x^2)^5(1+B(x))/(1-B(x^2))^3 + 2B(x)B(x^3)^2/(6(1-B(x^3))) + B(x)B(x^4)^2/(2(1-B(x^4))) + B(x)^2B(x^2)^2(7B(x)^2+5B(x)^3+3B(x^2)+B(x)(2+B(x^2)))/(2(1-B(x))(1-B(x^2))) + B(x)^5B(x^2)^2(3+2B(x))/((1-B(x))^2(1-B(x^2))) + B(x)^6B(x^2)^2/((1-B(x))^3(1-B(x^2))) + B(x)^2B(x^2)^4/((1-B(x))(1-B(x^2))^2) + B(x^2)B(x^4)^2(1+B(x))/(2(1-B(x^2))(1-B(x^4))), where B(x) is the generating function for rooted trees with n nodes in A000081.

A171860 Number of n-cell fixed polycubes that are proper in n-2 dimensions.

Original entry on oeis.org

0, 1, 17, 348, 8640, 254800, 8749056, 343901376, 15257600000, 755110160640, 41278242816000, 2471677136321536, 160961785787056128, 11330322120000000000, 857485369051342438400, 69444841895469240729600, 5993559601317659925282816, 549242871950650346384195584

Offset: 2

N. J. A. Sloane, Oct 16 2010

nonn

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
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.
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275. See Th. 6.

Vincenzo Librandi, Table of n, a(n) for n = 2..100
A. Asinowski, G. Barequet, R. Barequet, and G. Rote, Proper n-Cell Polycubes in n-3 Dimensions, J. Int. Seq. 15 (2012) #12.8.4.
M. Shalah, Formulae and growth rates of animals on cubical and triangular lattices, PhD Thesis, Israel Inst. Techn. (2017).

Cf. A127670, A191092, A036364 (free).

Diagonal 2 of A195739.

[2^(n-3)*n^(n-5)*(n-2)*(2*n^2-6*n+9): n in [2..20]]; // Vincenzo Librandi, May 26 2011

Table[2^(n-3)n^(n-5)(n-2)(2n^2-6n+9),{n,2,30}] (* Harvey P. Dale, Nov 27 2024 *)

a(n) = 2^(n-3)*n^(n-5)*(n-2)*(2*n^2 - 6*n + 9).

Slightly edited by Gill Barequet, May 25 2011

A036365 Number of chiral n-ominoes in n-2 space.

Original entry on oeis.org

0, 2, 6, 17, 49, 135, 361, 951, 2493, 6497, 16837, 43498, 112164, 288741, 742294, 1906552, 4893835, 12555662, 32201344, 82566738, 211675672, 542621858, 1390929877, 3565435302, 9139718572, 23430209922, 60069035611, 154014868677

Offset: 3

Robert A. Russell

a(n) is Lunnon's DR(n,n-2) - DE(n,n-2).

			0 chiral trominoes in 1-space;
2 pairs of chiral tetrominoes (L,S) in 2-space;
6 pairs of chiral pentominoes in 3-space.

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

Cf. A045648, A045649, A036364.

sc[ n_, k_ ] := sc[ n, k ]=c[ n+1-k, 1 ]+If[ n<2k, 0, sc[ n-k, k ](-1)^k ]; c[ 1, 1 ] := 1;
c[ n_, 1 ] := c[ n, 1 ]=Sum[ c[ i, 1 ]sc[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
c[ n_, k_ ] := c[ n, k ]=Sum[ c[ i, 1 ]c[ n-i, k-1 ], {i, 1, n-1} ];
Table[ c[ i, 3 ]/2+5c[ i, 4 ]/8+Sum[ c[ i, j ], {j, 5, i} ]+If[ OddQ[ i ], 0,
3c[ i/2, 2 ](-1)^(i/2)/8-If[ OddQ[ i/2 ], 0, c[ i/4, 1 ](-1)^(i/4)/4 ] ]
+Sum[ c[ j, 1 ](c[ i-2j, 1 ]/2+c[ i-2j, 2 ]/4)(-1)^j, {j, 1, (i-1)/2} ], {i, 3, 30} ]

G.f.: C^3(x)/2 + C(x)C(-x^2)/2 + 5C^4(x)/8 + C^2(x)C(-x^2)/4 + 3C^2(-x^2)/8 - C(-x^4)/4 + C^5(x)/(1-C(x)), where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled (that is, C(x) is the g.f. of the sequence A045648).

A036367 Number of free orthoplex n-ominoes with cell centers determining n-2 space.

Original entry on oeis.org

1, 2, 8, 25, 86, 272, 875, 2732, 8505, 26104, 79708, 241522, 728632, 2187951, 6548819, 19542662, 58184124, 172880565, 512837063, 1519158462, 4494920802, 13286473612, 39240530012, 115811180864, 341588823740, 1007007175952, 2967361180383

Offset: 4

Robert A. Russell

Orthoplex polyominoes are multidimensional polyominoes that do not extend more than two units along any axis.

			a(4)=1 because there is 1 tetromino (a square) in 2^2 space;
a(5)=2 because there are 2 pentominoes in 2^3 space;
a(6)=8 because in 2^4 space there are 8 hexominoes that have cell centers determining 4-space.

Cf. A000081, A036364.

sb[ n_, k_ ] := sb[ n, k ]=b[ n+1-k, 1 ]+If[ n<2k, 0, sb[ n-k, k ] ]; b[ 1, 1 ] := 1;
b[ n_, 1 ] := b[ n, 1 ]=Sum[ b[ i, 1 ]sb[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
b[ n_, k_ ] := b[ n, k ]=Sum[ b[ i, 1 ]b[ n-i, k-1 ], {i, 1, n-1} ];
Table[ b[ i, 4 ]/8+Sum[ b[ i, j ], {j, 5, i} ]/2+If[ OddQ[ i ], 0, 3b[ i/2, 2 ]/8
+If[ OddQ[ i/2 ], 0, b[ i/4, 1 ]/4 ]+Sum[ b[ i/2, j ], {j, 3, i/2} ]/2 ]
+Sum[ b[ j, 1 ]b[ i-2j, 2 ]/4+Sum[ If[ OddQ[ k ], b[ j,
(k-1)/2 ]b[ i-2j, 1 ], 0 ], {k, 5, i} ]/2, {j, 1, (i-1)/2} ], {i, 4, 30} ]

G.f.: (B^2(x) + B(x^2))^2/8 + B^2(x^2)/4 + B(x^4)/4 + B^5(x)/(2 - 2*B(x)) + (B(x) + B(x^2))*B^2(x^2)/(2 - 2*B(x^2)) where B(x) is the generating function for rooted trees with n nodes in A000081.

cp's OEIS Frontend

A049430 Triangle read by rows: T(n,d) is the number of distinct properly d-dimensional polyominoes (or polycubes) with n cells (n >= 1, d >= 0).

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A195738 Triangle read by rows: DR(n,d) is the number of properly d-dimensional polyominoes with n cells, modulo translations and rotations (n >= 1, 0 <= d <= n-1).

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A049429 Triangle T(n,d) = number of distinct d-dimensional polyominoes (or polycubes) with n cells (0 < d < n).

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A355053 Number of unoriented multidimensional n-ominoes with cell centers determining n-3 space.

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A171860 Number of n-cell fixed polycubes that are proper in n-2 dimensions.

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A036365 Number of chiral n-ominoes in n-2 space.

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A036367 Number of free orthoplex n-ominoes with cell centers determining n-2 space.

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