OFFSET
1,9
COMMENTS
The length here is the longer of the two dimensions.
LINKS
John Mason, Table of n, a(n) for n = 1..171 (first 18 rows)
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 1, 3, 1;
0, 0, 8, 3, 1;
0, 0, 8, 21, 5, 1;
0, 0, 7, 59, 36, 5, 1;
0, 0, 3, 137, 167, 54, 7, 1;
0, 0, 1, 223, 669, 307, 77, 7, 1;
0, 0, 0, 287, 2089, 1627, 539, 103, 9, 1;
0, 0, 0, 255, 5472, 7126, 3237, 839, 134, 9, 1;
0, 0, 0, 169, 11919, 27504, 16706, 5851, 1271, 168, 11, 1;
...
Illustration for n = 5:
The free polyominoes with five cells are also called free pentominoes.
For k = 1 there are no free pentominoes of length 1, so T(5,1) = 0.
For k = 2 there are no free pentominoes of length 2, so T(5,2) = 0.
For k = 3 there are eight free pentominoes of length 3 as shown below, so T(5,3) = 8.
_ _ _ _ _ _ _ _ _ _ _ _ _ _
|_|_| |_|_| _|_|_| |_|_|_| |_| |_|_ _|_|_ |_|_|
|_|_| |_|_ |_|_| |_| |_|_ _ |_|_|_ |_|_|_| |_|_
|_| |_|_| |_| |_| |_|_|_| |_|_| |_| |_|_|
.
For k = 4 there are three free pentominoes of length 4 as shown below, so T(5,4) = 3.
_ _ _
|_| _|_| _|_|
|_| |_|_| |_|_|
|_|_ |_| |_|
|_|_| |_| |_|
.
For k = 5 there is only one free pentomino of length 5 as shown below, so T(5,5) = 1.
_
|_|
|_|
|_|
|_|
|_|
.
Therefore the 5th row of the triangle is [0, 0, 8, 3, 1] and the row sum is A000105(5) = 12.
.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Jan 07 2025
EXTENSIONS
Terms a(37) and beyond from Jinyuan Wang, Jan 08 2025
STATUS
approved
