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

1,5

Comments

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

Links

R. J. Mathar, Table of n, a(n) for n = 1..109

Index entries for sequences related to polyominoes

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.

Example

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;

Crossrefs

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

Sequence in context: A110511 A082950 A060102 * A199065 A152237 A176282

Adjacent sequences: A308356 A308357 A308358 * A308360 A308361 A308362

Keyword

nonn,tabl

Author

R. J. Mathar, May 22 2019

Status

approved