%I #23 Oct 10 2022 10:18:06

%S 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,

%T 1,1,123,721,1027,638,190,24,1,1,228,1866,3468,2817,1226,275,28,1,1,

%U 421,4728,11132,11254,6391,2110,376,32,1,1,776,11804,34558,42099,29388,12758,3354,493,36,1

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

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

%H R. J. Mathar, <a href="/A308359/b308359.txt">Table of n, a(n) for n = 1..109</a>

%H <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>

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

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

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

%e 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.

%e The triangle starts

%e 1;

%e 1, 1;

%e 1, 4, 1;

%e 1, 9, 8, 1;

%e 1, 18, 31, 12, 1;

%e 1, 35, 95, 68, 16, 1;

%e 1, 66, 269, 282, 121, 20, 1;

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

%K nonn,tabl

%O 1,5

%A _R. J. Mathar_, May 22 2019