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

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

Offset

4,2

Comments

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.

Links

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

Formula

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.

Example

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.

Mathematica

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]

Crossrefs

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).

Sequence in context: A208599 A325733 A059625 * A023010 A303103 A258459

Adjacent sequences: A355050 A355051 A355052 * A355054 A355055 A355056

Keyword

nonn,easy

Author

Robert A. Russell, Jun 16 2022

Status

approved