A182797 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the k X k X k triangular grid.

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 24, 5, 0, 0, 6, 192, 60, 6, 0, 0, 6, 2112, 1620, 120, 7, 0, 0, 6, 32640, 98820, 7680, 210, 8, 0, 0, 6, 718080, 13638780, 1574400, 26250, 336, 9, 0, 0, 6, 22665216, 4260983940, 1034019840, 13676250, 72576, 504, 10

Offset

1,3

Comments

The k X k X k triangular grid has k rows with i vertices in row i. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has A000217(k) vertices and 3*A000217(k-1) edges altogether.

The coefficients of the chromatic polynomials for the column sequences are given by the rows of A193283. - Georg Fischer, Jul 31 2023

Links

Table of n, a(n) for n=1..55.

Wikipedia, Triangular grid graph

Wikipedia, Chromatic polynomial

Example

Square array A(n,k) begins:
  1,   0,    0,       0,          0,             0,  ...
  2,   0,    0,       0,          0,             0,  ...
  3,   6,    6,       6,          6,             6,  ...
  4,  24,  192,    2112,      32640,        718080,  ...
  5,  60, 1620,   98820,   13638780,    4260983940,  ...
  6, 120, 7680, 1574400, 1034019840, 2175789895680,  ...

Crossrefs

Columns k=1-11 give: A000027, A007531, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182795, A182796.

Rows n=1-10 give: A000007(k-1), A000038(k-1), A040006(k-1), A182798, A153467*4, A153468*5, A153469*6, A153470*7, A153471*8, A153472*9, A153473*10.

Cf. A000217, A193283.

Sequence in context: A136274 A290976 A114699 * A212163 A212195 A228926

Adjacent sequences: A182794 A182795 A182796 * A182798 A182799 A182800

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 02 2010

Status

approved