A308359 Triangle T(n,w) read by rows: the number of fixed polyominoes with n cells and width w of the convex hull.

1, 1, 1, 1, 4, 1, 1, 9, 8, 1, 1, 18, 31, 12, 1, 1, 35, 95, 68, 16, 1, 1, 66, 269, 282, 121, 20, 1, 1, 123, 721, 1027, 638, 190, 24, 1, 1, 228, 1866, 3468, 2817, 1226, 275, 28, 1, 1, 421, 4728, 11132, 11254, 6391, 2110, 376, 32, 1, 1, 776, 11804, 34558, 42099, 29388, 12758, 3354, 493, 36, 1

Offset: 1

Author

R. J. Mathar, May 22 2019

Keywords

• nonn
• tabl

Comments

The sequence counts the fixed n-ominoes with prescribed bounding box width w and variable height w <= h <= n.

Examples

			T(3,2) = 4 counts the 4 variants of the L-shaped tromino rotated by multiples of 90 degrees. T(4,2) = 9 counts one O-tetromino in a 2 X 2 box, 4 L-tetrominoes in a 3 X 2 box, 2 T-tetromoes in a 3 X 2 box, and 2 Z-tetrominoes in a 3 X 2 box.
The triangle starts
  1;
  1,   1;
  1,   4,   1;
  1,   9,   8,   1;
  1,  18,  31,  12,   1;
  1,  35,  95,  68,  16,   1;
  1,  66, 269, 282, 121,  20,   1;
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..109
• Index entries for sequences related to polyominoes

Crossrefs

Cf. A027053 (column w=2), A335606 (w=3), A001168 (row sums), A273895, A292357 (prescribed w and h).

Formula

T(n,1) = T(n,n) = 1 (the straight n-ominoes).

T(n,n-1) = 4*n-8 for n >= 3 (width n-1 and height 2).

Conjecture: T(n,n-2) = 8*n^2 - 51*n + 86 for n >= 5.