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A355997
Triangle read by rows: T(n,d) is the number of fixed orthoplex n-ominoes with cell centers determining d-space.
3
1, 0, 4, 0, 1, 32, 0, 0, 48, 400, 0, 0, 28, 1728, 6912, 0, 0, 8, 4240, 62720, 153664, 0, 0, 1, 7272, 344320, 2457600, 4194304, 0, 0, 0, 8720, 1465600, 23872320, 105815808, 136048896, 0, 0, 0, 7136, 5254576, 182691200, 1603840000, 5017600000, 512000000
OFFSET
2,3
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.
Conjecture: T(n,n-4) = 2^(n-7) * n^(n-9) * (n-4) * (n-5) * (n-6) * (n^6-14*n^5+65*n^4-189*n^3+594*n^2-1305*n+6832) / 6 ~ A259015(n) / 8.
LINKS
EXAMPLE
Triangle begins with T(2,1):
n\d 1 2 3 4 5 6 7 8 9
2 1
3 0 4
4 0 1 32
5 0 0 48 400
6 0 0 28 1728 6912
7 0 0 8 4240 62720 153664
8 0 0 1 7272 344320 2457600 4194304
9 0 0 0 8720 1465600 23872320 105815808 136048896
10 0 0 0 7136 5254576 182691200 1603840000 5017600000 512000000
CROSSREFS
Cf. A195739 (multidimensional).
Diagonals (with formulas) are A127670, A355998, A355999.
Sequence in context: A334703 A259938 A342202 * A136452 A247703 A280639
KEYWORD
nonn,tabl
AUTHOR
Robert A. Russell, Jul 22 2022
STATUS
approved