A384782 Triangle read by rows: T(n,k) is the number of face-connected polyhedral components consisting of k cuboctahedra and n-k octahedra in the rectified cubic honeycomb up to translation, rotation, and reflection of the honeycomb, 0<=k<=n.

1, 1, 1, 0, 1, 1, 0, 3, 4, 2, 0, 3, 18, 12, 7, 0, 6, 60, 126, 75, 23, 0, 3, 165, 751, 1025, 473, 112, 0, 3, 346, 3784, 9414, 8936, 3539, 607, 0, 1, 565, 14112, 66503, 108739, 80531, 27027, 3811, 0, 1, 723, 42420, 362939, 994542, 1204093, 725795, 212122, 25413, 0, 0, 723, 101237, 1586479, 7065791, 13389295, 12792264, 6512671, 1678783, 178083

Offset

0,8

Comments

Also the number of face-connected polyhedral components consisting of k truncated cubes and n-k octahedra in the truncated cubic honeycomb up to translation, rotation, and reflection of the honeycomb.

Row sums are given by A384254.

Links

Table of n, a(n) for n=0..65.

Peter Kagey, Illustration of row 4.

Wikipedia, Convex uniform honeycomb

Formula

T(n,n) = A038119(n).

Example

Table begins:
   0 | 1;
   1 | 1, 1;
   2 | 0, 1,   1;
   3 | 0, 3,   4,     2;
   4 | 0, 3,  18,    12,      7;
   5 | 0, 6,  60,   126,     75,     23;
   6 | 0, 3, 165,   751,   1025,    473,     112;
   7 | 0, 3, 346,  3784,   9414,   8936,    3539,    607;
   8 | 0, 1, 565, 14112,  66503, 108739,   80531,  27027,   3811;
   9 | 0, 1, 723, 42420, 362939, 994542, 1204093, 725795, 212122, 25413;

Crossrefs

Cf. A384254.

Cf. A365970 (tetrahedral-octahedral honeycomb), A384486 (quarter cubic honeycomb), A384755 (omnitruncated cubic honeycomb).

Sequence in context: A352770 A019829 A383115 * A200125 A091528 A236679

Adjacent sequences: A384779 A384780 A384781 * A384783 A384784 A384785

Keyword

nonn,tabl

Author

Peter Kagey and Bert Dobbelaere, Jun 09 2025

Status

approved