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A308359
Triangle T(n,w) read by rows: the number of fixed polyominoes with n cells and width w of the convex hull.
6
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.
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
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, May 22 2019
STATUS
approved