A355048 Number of unoriented orthoplex n-ominoes with cell centers determining n-3 space.

3, 18, 122, 655, 3240, 14531, 61520, 247381, 958434, 3598594, 13180348, 47274577, 166642096, 578750970, 1984671466, 6731351834, 22612409886, 75321920403, 249028297179, 817867225710, 2670093233760, 8670380548402

Offset

6,1

Comments

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. For unoriented polyominoes, chiral pairs are counted as one.

Links

Robert A. Russell, Table of n, a(n) for n = 6..100

Robert A. Russell, Trunk Generating Functions

Formula

a(n) = A355047(n) - A355049(n) = (A355047(n) + A355050(n)) / 2 = A355049(n) + A355050(n).

G.f.: (14B(x}^6 + 3B(x}^7 + 6B(x}^4B(x^2} + 6B(x}^5B(x^2} + 18B(x}^2B(x^2}^2 + 3B(x}^3B(x^2}^2 + 26B(x^2}^3 + 6 B(x}B(x^2}(B(x^2}^2 + B(x^4}) + 4B(x^3}^2 + 4B(x^6}) / 24 + B(x}^3 (38B(x}^4 + 9B(x}^5 + 4B(x}^2B(x^2} + 10B(x}^3B(x^2} + 2B(x^2}^2 + B(x}B(x^2}^2) / (8(1-B(x})) + B(x}^6 (16B(x}^2 + 6B(x}^3 + B(x^2} + B(x} (5 + 2B(x^2})) / (2(1-B(x})^2) + B(x}^7 (2 + 42B(x} + 51B(x}^2 + 24B(x}^3 + 3B(x^2}) / (12(1-B(x})^3) + B(x}^9 (17 + 8B(x}) / (8(1-B(x})^4) + 3B(x}^10 / (8(1-B(x})^5) + B(x^2}^2(B(x}^4 + 4B(x}^2 B(x^2} + 12B(x^2}^2 + B(x^4} + B(x} (8B(x^2} + 5B(x^2}^2 + B(x^4})) / (4(1-B(x^2})) + B(x^2}^4 (8 + 16B(x^2} + B(x} (19 + 8B(x^2})) / (8(1-B(x^2})^2) + 3(1 + B(x})B(x^2}^5 / (4(1-B(x^2})^3) + 2B(x}B(x^3}^2 / (6(1-B(x^3})) + B(x}B(x^4}^2 / (4(1-B(x^4})) + B(x}^2B(x^2}^2(5B(x}^3 + 2B(x^2} + B(x}(2 + B(x^2})) / (4(1-B(x})(1-B(x^2})) + B(x}^5(1+4B(x})B(x^2}^2 / (4(1-B(x})^2(1-B(x^2})) + B(x}^6 B(x^2}^2 / (4(1-B(x})^3(1-B(x^2})) + 3B(x}^2B(x^2}^4 / (8(1-B(x})(1-B(x^2})^2) + B(x^2}(1+B(x})B(x^4}^2 / (4(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(6)=3 because there are 3 hexominoes in 2^3 space. The two vacant cells share just a face, an edge, or a vertex.

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[(14B[x]^6 + 3B[x]^7 + 6B[x]^4B[x^2] + 6B[x]^5B[x^2] + 18B[x]^2B[x^2]^2 + 3B[x]^3B[x^2]^2 + 26B[x^2]^3 + 6 B[x]B[x^2](B[x^2]^2 + B[x^4]) + 4B[x^3]^2 + 4B[x^6]) / 24 + B[x]^3 (38B[x]^4 + 9B[x]^5 + 4B[x]^2B[x^2] + 10B[x]^3B[x^2] + 2B[x^2]^2 + B[x]B[x^2]^2) / (8(1-B[x])) + B[x]^6 (16B[x]^2 + 6B[x]^3 + B[x^2] + B[x] (5 + 2B[x^2])) / (2(1-B[x])^2) + B[x]^7 (2 + 42B[x] + 51B[x]^2 + 24B[x]^3 + 3B[x^2]) / (12(1-B[x])^3) + B[x]^9 (17 + 8B[x]) / (8(1-B[x])^4) + 3B[x]^10 / (8(1-B[x])^5) + B[x^2]^2(B[x]^4 + 4B[x]^2 B[x^2] + 12B[x^2]^2 + B[x^4] + B[x] (8B[x^2] + 5B[x^2]^2 + B[x^4])) / (4(1-B[x^2])) + B[x^2]^4 (8 + 16B[x^2] + B[x] (19 + 8B[x^2])) / (8(1-B[x^2])^2) + 3(1 + B[x])B[x^2]^5 / (4(1-B[x^2])^3) + 2B[x]B[x^3]^2 / (6(1-B[x^3])) + B[x]B[x^4]^2 / (4(1-B[x^4])) + B[x]^2B[x^2]^2(5B[x]^3 + 2B[x^2] + B[x](2 + B[x^2])) / (4(1-B[x])(1-B[x^2])) + B[x]^5(1+4B[x])B[x^2]^2 / (4(1-B[x])^2(1-B[x^2])) + B[x]^6 B[x^2]^2 / (4(1-B[x])^3(1-B[x^2])) + 3B[x]^2B[x^2]^4 / (8(1-B[x])(1-B[x^2])^2) + B[x^2](1+B[x])B[x^4]^2 / (4(1-B[x^2])(1-B[x^4])), {x, 0, nmax}], x], 6]

Crossrefs

Cf. A355047 (oriented), A355049 (chiral), A355050 (achiral) A355051 (asymmetric), A000081 (rooted trees).

Other dimensions: A036367 (n-2), A000055 (n-1), A355053 (multidimensional).

Sequence in context: A295537 A183145 A127188 * A074558 A074564 A108241

Adjacent sequences: A355045 A355046 A355047 * A355049 A355050 A355051

Keyword

nonn,easy

Author

Robert A. Russell, Jun 16 2022

Status

approved