A191092 -id:A191092 - cp's OEIS frontend

A127670 Discriminants of Chebyshev S-polynomials A049310.

1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104, 75613185918270483380568064

Offset: 1

Wolfdieter Lang, Jan 23 2007

a(n-1) is the number of fixed n-cell polycubes that are proper in n-1 dimensions (Barequet et al., 2010).
From Rigoberto Florez, Sep 02 2018: (Start)
a(n-1) is the discriminant of the Morgan-Voyce Fibonacci-type polynomial B(n).
Morgan-Voyce Fibonacci-type polynomials are defined as B(0) = 0, B(1) = 1 and B(n) = (x+2)*B(n-1) - B(n-2) for n > 1.
The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n-1).
Fibonacci polynomials are defined as F(0) = 0, F(1) = 1 and F(n) = x*F(n-1) + F(n-2) for n > 1. (End)
The first 6 values are the dimensions of the polynomial ring in 3n variables xi, yi, zi for 1 <= i <= n modulo the ideal generated by x1^a y1^b z1^c + ... + xn^a yn^b zn^c for 0 < a+b+c <= n (see Fact 2.8.1 in Haiman's paper). - Mike Zabrocki, Dec 31 2019

			n=3: The zeros are [sqrt(2),0,-sqrt(2)]. The Vn(xn[1],...,xn[n]) matrix is [[1,1,1],[sqrt(2),0,-sqrt(2)],[2,0,2]]. The squared determinant is 32 = a(3). - _Wolfdieter Lang_, Aug 07 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.
G. Barequet and 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.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219 for T and U polynomials.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Andrei Asinowski, Gill Barequet, Ronnie Barequet, and Gunter Rote, Proper n-Cell Polycubes in n - 3 Dimensions, Journal of Integer Sequences, Vol. 15 (2012), #12.8.4.
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012. See Th. 1. [From _N. J. A. Sloane_, Oct 16 2010]
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, preprint, 1993.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17--76.
O. Khorunzhiy, Enumeration of tree-type diagrams assembled from oriented chains of edges, arXiv:2207.00766 [math.CO], 2022.
Andrew Snowden, Measures for the colored circle, arXiv:2302.08699 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 dimensions, resp.).
Cf. A243953, A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.
A317403 is essentially the same sequence.
Diagonal 1 of A195739.

[((n+1)^n/(n+1)^2)*2^n: n in [1..20]]; // Vincenzo Librandi, Jun 23 2014

Table[((n + 1)^n)/(n + 1)^2 2^n, {n, 1, 30}] (* Vincenzo Librandi, Jun 23 2014 *)

a(n) = ((n+1)^(n-2))*2^n, n >= 1.
a(n) = (Det(Vn(xn[1],...,xn[n])))^2 with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=2*cos(Pi*i/(n+1)), i=1..n, are the zeros of S(n,x):=U(n,x/2).
a(n) = ((-1)^(n*(n-1)/2))*Product_{j=1..n} ((d/dx)S(n,x)|_{x=xn[j]}), n >= 1, with the zeros xn[j], j=1..n, given above.
a(n) = A007830(n-2)*A000079(n), n >= 2. - Omar E. Pol, Aug 27 2011
E.g.f.: -LambertW(-2*x)*(2+LambertW(-2*x))/(4*x). - Vaclav Kotesovec, Jun 22 2014

Slightly edited by Gill Barequet, May 24 2011

A195739 Triangle read by rows: DX(n,d) = number of properly d-dimensional polyominoes with n cells, modulo translations (n>=1, 0 <= d <= n-1).

Original entry on oeis.org

1, 0, 1, 0, 1, 4, 0, 1, 17, 32, 0, 1, 61, 348, 400, 0, 1, 214, 2836, 8640, 6912, 0, 1, 758, 21225, 129288, 254800, 153664, 0, 1, 2723, 154741, 1688424, 6160640, 8749056, 4194304, 0, 1, 9908, 1123143, 20762073, 125055400, 313921008, 343901376, 136048896

Offset: 1

N. J. A. Sloane, Sep 23 2011

According to Barequet-Barequet-Rote, p. 261, the value DX(7, 6) = 134209 given by W. F. Lunnon is incorrect; it should be 153664, see A127670. - Alexander Knapp, May 13 2013

			Triangle begins with DX(1,0):
n\d 0  1   2     3      4      5      6
---------------------------------------
1...1
2...0  1
3...0  1   4
4...0  1  17    32
5...0  1  61   348    400
6...0  1 214  2836   8640   6912
7...0  1 758 21225 129288 254800 153664
...

Robert A. Russell, Table of n, a(n) for n = 1..60
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.
W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (1975), no. 4, pp. 366-367.

Columns give A006762, A006763, A006764. Cf. A195738, A049430.

Diagonals (with formulas) are A127670, A171860, A191092, A259015, A290738.

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

A355054 Number of chiral pairs of multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

6, 54, 297, 1341, 5468, 20519, 72168, 242886, 791780, 2514453, 7814225, 23863941, 71835845, 213601046, 628450974, 1832227629, 5299559865, 15221688836, 43450246045, 123345029035, 348417524877, 979803281560

Offset: 5

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). Each member of a chiral pair is a reflection but not a rotation of the other.

			a(5)=6 because there are 6 chiral pairs of pentominoes in 2-space.

Robert A. Russell, Table of n, a(n) for n = 5..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), A355053 (unoriented), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A045648 (rooted chiral), A195738 (Lunnon's DR), A049430 (Lunnon's DE).

Other dimensions: A036365 (n-2), A045649 (n-1), A355049 (orthoplex).

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}];
nmax = 30; K[x_] := Sum[c[i,1] x^i, {i,0,nmax}]
Drop[CoefficientList[Series[(12 K[x]^4 + 87 K[x]^5 + 50 K[x]^6 + 3 K[x]^7 + 18 K[x]^3 K[-x^2] + 36 K[x]^4 K[-x^2] + 6 K[x]^5 K[-x^2] - 12 K[-x^2]^2 - 27 K[x] K[-x^2]^2 - 6 K[x]^2 K[-x^2]^2 + 3 K[x]^3 K[-x^2]^2 - 16 K[-x^2]^3 - 6 K[x] K[-x^2]^3 + 4 K[x^3]^2 - 6 K[x] K[-x^4] - 6 K[x] K[-x^2] K[-x^4] + 4 K[-x^6]) / 24 + K[x]^2 (16 K[x]^3 + 159 K[x]^4 + 112 K[x]^5 + 9 K[x]^6 + 14 K[x]^2 K[-x^2] + 32 K[x]^3 K[-x^2] + 10 K[x]^4 K[-x^2] - K[-x^2]^2 + K[x]^2 K[-x^2]^2) / (8 (1-K[x])) + K[x]^5 (2 K[x] + 67 K[x]^2 + 46 K[x]^3 + 6 K[x]^4 + 3 K[-x^2] + 6 K[x] K[-x^2] + 2 K[x]^2 K[-x^2]) / (2 (1-K[x])^2) - K[-x^2] (2 K[x]^2 K[-x^2] + 7 K[-x^2]^2 + 17 K[x] K[-x^2]^2 + 2 K[x]^2 K[-x^2]^2 + 7 K[-x^2]^3 + 5 K[x] K[-x^2]^3 + K[-x^4] + K[x] K[-x^4] + K[-x^2] K[-x^4] + K[x] K[-x^2] K[-x^4]) / (4 (1-K[-x^2])) + K[x]^6 (4 K[x] + 153 K[x]^2 + 75 K[x]^3 + 12 K[x]^4 + 3 K[-x^2] + 3 K[x] K[-x^2]) / (6 (1-K[x])^3) - K[x]^2 K[-x^2]^2 (K[x] + K[-x^2]) / ((1-K[x]) (1-K[-x^2])) + (K[x] K[x^3]^2) / (3 (1-K[x^3])) + K[x]^9 (21 + 4 K[x]) / (2 (1-K[x])^4) - K[-x^2]^4 (6 + 7 K[x] + 2 K[-x^2] + 2 K[x] K[-x^2]) / (2 (1-K[-x^2])^2) + 3 K[x]^10 / (2 (1-K[x])^5) - K[x]^2 K[-x^2]^4 / (2 (1-K[x]) (1-K[-x^2])^2) - (1 + K[x]) K[-x^2]^5 / (1-K[-x^2])^3, {x,0,nmax}], x], 5]

a(n) = A355052(n) - A355053(n) = (A355052(n) - A355055(n)) / 2 = A355053(n) - A355055(n).
a(n) = A195738(n,n-3) - A049430(n,n-3), diagonals of Lunnon's DR and DE arrays.
G.f.: (12 C(x)^4 + 87 C(x)^5 + 50 C(x)^6 + 3 C(x)^7 + 18 C(x)^3 C(-x^2) + 36 C(x)^4 C(-x^2) + 6 C(x)^5 C(-x^2) - 12 C(-x^2)^2 - 27 C(x) C(-x^2)^2 - 6 C(x)^2 C(-x^2)^2 + 3 C(x)^3 C(-x^2)^2 - 16 C(-x^2)^3 - 6 C(x) C(-x^2)^3 + 4 C(x^3)^2 - 6 C(x) C(-x^4) - 6 C(x) C(-x^2) C(-x^4) + 4 C(-x^6)) / 24 + C(x)^2 (16 C(x)^3 + 159 C(x)^4 + 112 C(x)^5 + 9 C(x)^6 + 14 C(x)^2 C(-x^2) + 32 C(x)^3 C(-x^2) + 10 C(x)^4 C(-x^2) - C(-x^2)^2 + C(x)^2 C(-x^2)^2) / (8 (1-C(x))) + C(x)^5 (2 C(x) + 67 C(x)^2 + 46 C(x)^3 + 6 C(x)^4 + 3 C(-x^2) + 6 C(x) C(-x^2) + 2 C(x)^2 C(-x^2)) / (2 (1-C(x))^2) - C(-x^2) (2 C(x)^2 C(-x^2) + 7 C(-x^2)^2 + 17 C(x) C(-x^2)^2 + 2 C(x)^2 C(-x^2)^2 + 7 C(-x^2)^3 + 5 C(x) C(-x^2)^3 + C(-x^4) + C(x) C(-x^4) + C(-x^2) C(-x^4) + C(x) C(-x^2) C(-x^4)) / (4 (1-C(-x^2))) + C(x)^6 (4 C(x) + 153 C(x)^2 + 75 C(x)^3 + 12 C(x)^4 + 3 C(-x^2) + 3 C(x) C(-x^2)) / (6 (1-C(x))^3) - C(x)^2 C(-x^2)^2 (C(x) + C(-x^2)) / ((1-C(x)) (1-C(-x^2))) + (C(x) C(x^3)^2) / (3 (1-C(x^3))) + C(x)^9 (21 + 4 C(x)) / (2 (1-C(x))^4) - C(-x^2)^4 (6 + 7 C(x) + 2 C(-x^2) + 2 C(x) C(-x^2)) / (2 (1-C(-x^2))^2) + 3 C(x)^10 / (2 (1-C(x))^5) - C(x)^2 C(-x^2)^4 / (2 (1-C(x)) (1-C(-x^2))^2) - (1 + C(x)) C(-x^2)^5 / (1-C(-x^2))^3 where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

A355056 Number of asymmetric multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

5, 46, 275, 1283, 5281, 19607, 68476, 227196, 727780, 2263148, 6881482, 20529511, 60312548, 174870492, 501443277, 1424142358, 4011274417, 11216074419, 31160837273, 86078096135, 236568911194, 647181951619

Offset: 5

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). An asymmetric polyomino has a symmetry group of order 1.

			a(5)=5 as there are exactly five asymmetric pentominoes in 2-space.

Robert A. Russell, Table of n, a(n) for n = 5..100
Robert A. Russell, Trunk Generating Functions

Cf. A355052 (oriented), A355053 (unoriented), A355054 (chiral), A355055 (achiral), A191092 (fixed), A004111 (rooted asymmetric).

Other dimensions: A036366 (n-2), A000220 (n-1), A355051 (orthoplex).

sa[n_, k_] := sa[n, k] = a[n+1-k, 1] + If[n < 2 k, 0, -sa[n-k, k]];
a[1, 1] := 1; a[n_, 1] := a[n, 1] = Sum[a[i, 1] sa[n-1, i] i, {i, 1, n-1}]/(n-1);
a[n_, k_] := a[n, k] = Sum[a[i, 1] a[n-i, k-1], {i, 1, n-1}];
nmax = 30; A[x_] := Sum[a[i, 1] x^i, {i, 0, nmax}]
Drop[CoefficientList[Series[(4 A[x]^4 + 37 A[x]^5 + 12 A[x]^6 - 6 A[x]^3 A[x^2] - 10 A[x]^4 A[x^2] - 4 A[x^2]^2 - 17 A[x] A[x^2]^2 - 2 A[x^2]^3 + 2 A[x] A[x^4]) / 8 + (24 A[x]^5 + 515 A[x]^6 + 325 A[x]^7 + 24 A[x]^8 - 48 A[x]^4 A[x^2] - 96 A[x]^5 A[x^2] - 24 A[x]^6 A[x^2] - 21 A[x]^2 A[x^2]^2 + 21 A[x]^3 A[x^2]^2 - 14 A[x^2]^3 + 8 A[x] A[x^2]^3 + 6 A[x]^2 A[x^2]^3 + 4 A[x^3]^2 - 4 A[x] A[x^3]^2 + 24 A[x^2] A[x^4] - 18 A[x] A[x^2] A[x^4] - 6 A[x]^2 A[x^2] A[x^4] - 4 A[x^6] + 4 A[x] A[x^6]) / (24 (1-A[x])) + A[x]^5 (2 A[x] + 67 A[x]^2 + 46 A[x]^3 + 6 A[x]^4 - 3 A[x^2] - 6 A[x] A[x^2] - 2 A[x]^2 A[x^2]) / (2 (1-A[x])^2) - A[x^2] (2 A[x]^2 A[x^2] + 6 A[x]^3 A[x^2] + 2 A[x]^4 A[x^2] + 13 A[x^2]^2 + 31 A[x] A[x^2]^2 + 2 A[x]^2 A[x^2]^2 + 15 A[x^2]^3 + 5 A[x] A[x^2]^3 - 3 A[x^4] - 5 A[x] A[x^4] - 3 A[x^2] A[x^4] - A[x] A[x^2] A[x^4]) / (4 (1-A[x^2])) + A[x]^6 (4 A[x] + 153 A[x]^2 + 75 A[x]^3 + 12 A[x]^4 - 3 A[x^2] - 3 A[x] A[x^2]) / (6 (1-A[x])^3) - A[x]^2 A[x^2]^2 (2 A[x] + 7 A[x]^2 + 5 A[x]^3 + A[x^2] - A[x] A[x^2]) / (2 (1-A[x]) (1-A[x^2])) + A[x] A[x^3]^2 / (1-A[x^3]) / 3 + A[x]^9 (21 + 4 A[x]) / (2 (1-A[x])^4) - A[x]^5 (3 + 2 A[x]) A[x^2]^2 / ((1-A[x])^2 (1-A[x^2])) - A[x^2]^4 (5 + 7 A[x] + 3 A[x^2] + A[x] A[x^2]) / (1-A[x^2])^2 + A[x] A[x^4]^2 / (2 (1-A[x^4])) + 3 A[x]^10 / (2 (1-A[x])^5) - A[x]^6 A[x^2]^2 / ((1-A[x])^3 (1-A[x^2])) - 2 (1 + A[x]) A[x^2]^5 / (1-A[x^2])^3 + 3 (1 + A[x]) A[x^2] A[x^4]^2 / (2 (1-A[x^2]) (1-A[x^4])), {x,0,nmax}], x], 5]

G.f.: (4 A(x)^4 + 37 A(x)^5 + 12 A(x)^6 - 6 A(x)^3 A(x^2) - 10 A(x)^4 A(x^2) - 4 A(x^2)^2 - 17 A(x) A(x^2)^2 - 2 A(x^2)^3 + 2 A(x) A(x^4)) / 8 + (24 A(x)^5 + 515 A(x)^6 + 325 A(x)^7 + 24 A(x)^8 - 48 A(x)^4 A(x^2) - 96 A(x)^5 A(x^2) - 24 A(x)^6 A(x^2) - 21 A(x)^2 A(x^2)^2 + 21 A(x)^3 A(x^2)^2 - 14 A(x^2)^3 + 8 A(x) A(x^2)^3 + 6 A(x)^2 A(x^2)^3 + 4 A(x^3)^2 - 4 A(x) A(x^3)^2 + 24 A(x^2) A(x^4) - 18 A(x) A(x^2) A(x^4) - 6 A(x)^2 A(x^2) A(x^4) - 4 A(x^6) + 4 A(x) A(x^6)) / (24 (1-A(x))) + A(x)^5 (2 A(x) + 67 A(x)^2 + 46 A(x)^3 + 6 A(x)^4 - 3 A(x^2) - 6 A(x) A(x^2) - 2 A(x)^2 A(x^2)) / (2 (1-A(x))^2) - A(x^2) (2 A(x)^2 A(x^2) + 6 A(x)^3 A(x^2) + 2 A(x)^4 A(x^2) + 13 A(x^2)^2 + 31 A(x) A(x^2)^2 + 2 A(x)^2 A(x^2)^2 + 15 A(x^2)^3 + 5 A(x) A(x^2)^3 - 3 A(x^4) - 5 A(x) A(x^4) - 3 A(x^2) A(x^4) - A(x) A(x^2) A(x^4)) / (4 (1-A(x^2))) + A(x)^6 (4 A(x) + 153 A(x)^2 + 75 A(x)^3 + 12 A(x)^4 - 3 A(x^2) - 3 A(x) A(x^2)) / (6 (1-A(x))^3) - A(x)^2 A(x^2)^2 (2 A(x) + 7 A(x)^2 + 5 A(x)^3 + A(x^2) - A(x) A(x^2)) / (2 (1-A(x)) (1-A(x^2))) + A(x) A(x^3)^2 / (1-A(x^3)) / 3 + A(x)^9 (21 + 4 A(x)) / (2 (1-A(x))^4) - A(x)^5 (3 + 2 A(x)) A(x^2)^2 / ((1-A(x))^2 (1-A(x^2))) - A(x^2)^4 (5 + 7 A(x) + 3 A(x^2) + A(x) A(x^2)) / (1-A(x^2))^2 + A(x) A(x^4)^2 / (2 (1-A(x^4))) + 3 A(x)^10 / (2 (1-A(x))^5) - A(x)^6 A(x^2)^2 / ((1-A(x))^3 (1-A(x^2))) - 2 (1 + A(x)) A(x^2)^5 / (1-A(x^2))^3 + 3 (1 + A(x)) A(x^2) A(x^4)^2 / (2 (1-A(x^2)) (1-A(x^4))) where A(x) is the generating function for rooted identity trees with n nodes in A004111.

A355052 Number of oriented multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

1, 17, 131, 709, 3350, 14337, 57507, 218746, 803384, 2870707, 10044838, 34548917, 117224825, 393290329, 1307200931, 4310348599, 14116544717, 45959805027, 148860350902, 479938536114, 1541025955958, 4929773150983

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). This sequence is obtained using the first formula below. For oriented polyominoes, chiral pairs are counted as two.

			a(4)=1 because there is just one tetromino (with four cells aligned) in 1-space. a(5)=17 because there are 5 achiral and 6 chiral pairs of pentominoes in 2-space, excluding the 1-D straight pentomino.

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

Cf. A355053 (unoriented), A355054 (chiral), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A355047 (orthoplex), A195738 (Lunnon's DR).

a(n) = A355053(n) + A355054(n) = 2*A355053(n) - A355055(n) = 2*A355054(n) + A355055(n).

a(n) = A195738(n,n-3), the third diagonal of Lunnon's DR array.

A355055 Number of achiral multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

1, 5, 23, 115, 668, 3401, 16469, 74410, 317612, 1287147, 5015932, 18920467, 69496943, 249618639, 879998839, 3053446651, 10452089459, 35360685297, 118416973230, 393038044024, 1294335897888, 4232938101229, 13757913332396

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). This sequence is obtained using the first formula below. An achiral polyomino is identical to its reflection.

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

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

Cf. A355052 (oriented), A355053 (unoriented), A355054 (chiral), A355056 (asymmetric), A191092 (fixed), A355050 (orthoplex), A195738 (Lunnon's DR), A049430 (Lunnon's DE).

a(n) = A355053(n) - A355054(n) = 2*A355053(n) - A355052(n) = A355052(n) - 2*A355054(n).

a(n) = 2*A049430(n,n-3) - A195738(n,n-3), Lunnon's DE and DR arrays.

A355999 Number of fixed orthoplex n-ominoes with cell centers determining (n-3)-space.

Original entry on oeis.org

28, 4240, 344320, 23872320, 1603840000, 109616815616, 7785535242240, 580217967114240, 45559682696478700, 3774254616000000000, 329816052160897000000, 30372942170151000000000, 2943608844201080000000000

Offset: 6

Robert A. Russell, Jul 22 2022

nonn

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.

			For a(6)=28, 6 of the 8 cubes in the 2^3 space are used. There are 12 cases where the 2 empty cubes share a face, 12 cases where they share an edge, and 4 cases where they share a vertex.

Cf. A191092 (multidimensional), A355048 (unoriented), A355049 (chiral), A355051 (asymmetric).

Diagonal 3 of A355997.

Table[2^(n-6) n^(n-7) (n-3) (n-4) (n-5) (3n^3-17n^2+21n-78), {n,6,30}]

def A355999(n): return int(((1<Chai Wah Wu, Jul 26 2022

a(n) = 2^(n-6) * n^(n-7) * (n-3) * (n-4) * (n-5) * (3n^3-17n^2+21n-78) / 3.

a(n) ~ A191092(n) / 4.

cp's OEIS Frontend

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A195739 Triangle read by rows: DX(n,d) = number of properly d-dimensional polyominoes with n cells, modulo translations (n>=1, 0 <= d <= n-1).

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

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A355054 Number of chiral pairs of multidimensional n-ominoes with cell centers determining n-3 space.

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A355056 Number of asymmetric multidimensional n-ominoes with cell centers determining n-3 space.

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A355052 Number of oriented multidimensional n-ominoes with cell centers determining n-3 space.

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A355055 Number of achiral multidimensional n-ominoes with cell centers determining n-3 space.

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