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

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

Offset: 6

Author

Robert A. Russell, Jul 22 2022

Keywords

• nonn

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. Two fixed polyominoes are identical only if one is a translation of the other.

Examples

			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.
		

Crossrefs

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

Diagonal 3 of A355997.

Programs

Mathematica

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

Python

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

Formula

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.