A365970 Triangle read by rows: T(n,k) is the number of generalized polyforms on the tetrahedral-octahedral honeycomb with n cells, k of which are octahedra; 0 <= k <= n.

1, 1, 1, 0, 1, 0, 0, 3, 1, 0, 0, 3, 5, 1, 0, 0, 6, 24, 13, 1, 0, 0, 3, 74, 105, 13, 0, 0, 0, 3, 169, 727, 276, 11, 0, 0, 0, 1, 285, 3223, 3440, 432, 4, 0, 0, 0, 1, 356, 10853, 27632, 10141, 459, 2, 0, 0, 0, 0, 344, 27198, 155524, 134527, 19597, 314, 0, 0, 0

Offset

0,8

Comments

Polyforms are "free" in that they are counted up to rotation and reflection.

Conjecture: Columns and antidiagonals are unimodal.

Rows sums are given by A343909.

Links

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

Peter Kagey, Animation of the A343909(4) = 9 polyforms with 4 cells and T(4,1) = 3, T(4,2) = 5, and T(4,3) = 1 octahedra.

Math Stack Exchange, Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb.

Peter Kagey, Haskell program.

Formula

T(n,k) = 0 for k > n - floor((n - 1)/4).

Example

Triangle begins:
  1;
  1, 1;
  0, 1,   0;
  0, 3,   1,     0;
  0, 3,   5,     1,      0;
  0, 6,  24,    13,      1,      0;
  0, 3,  74,   105,     13,      0,     0;
  0, 3, 169,   727,    276,     11,     0,   0;
  0, 1, 285,  3223,   3440,    432,     4,   0, 0;
  0, 1, 356, 10853,  27632,  10141,   459,   2, 0, 0;
  0, 0, 344, 27198, 155524, 134527, 19597, 314, 0, 0, 0.

Crossrefs

Cf. A343909.

Sequence in context: A345371 A306268 A354490 * A144357 A122848 A272481

Adjacent sequences: A365967 A365968 A365969 * A365971 A365972 A365973

Keyword

nonn,tabl,hard

Author

Peter Kagey, Sep 23 2023

Status

approved