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

28, 4240, 344320, 23872320, 1603840000, 109616815616, 7785535242240, 580217967114240, 45559682696478720, 3774254616000000000, 329816052160897482752, 30372942170150997114880, 2943608844201076992245760, 299755360278739966554275840, 32020484430233600000000000000, 3582112654445477385157915705344

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

Links

Table of n, a(n) for n=6..21.

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.

Example

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.

Mathematica

Table[2^(n-6) n^(n-7) (n-3) (n-4) (n-5) (3n^3-17n^2+21n-78)/3, {n, 6, 30}]  (* corrected by Falk Hüffner, May 23 2026 *)

Prog

(Python)
def A355999(n): return int(((1<<n-6)*n**(n-7)*(n*(n*(n*(n*(n*(3*n - 53) + 366) - 1309) + 2943) - 4926) + 4680))//3) # Chai Wah Wu, Jul 26 2022

Crossrefs

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

Diagonal 3 of A355997.

Sequence in context: A290214 A036525 A364516 * A193985 A262018 A131315

Adjacent sequences: A355996 A355997 A355998 * A356000 A356001 A356002

Keyword

nonn

Author

Robert A. Russell, Jul 22 2022

Extensions

a(14) and a(16)-a(18) corrected by Falk Hüffner, May 23 2026

More terms from Michel Marcus, May 23 2026

Status

approved