OFFSET
1,6
COMMENTS
The width here is the shorter of the two dimensions.
LINKS
John Mason, Table of n, a(n) for n = 1..90 (first 18 rows)
EXAMPLE
Triangle begins:
1;
1;
1, 1;
1, 4;
1, 5, 6;
1, 12, 22;
1, 18, 71, 18;
1, 37, 193, 138;
1, 60, 490, 661, 73;
1, 117, 1221, 2547, 769;
1, 200, 3011, 8417, 5189, 255;
1, 379, 7393, 26164, 25920, 3743;
1, 669, 18025, 78074, 108834, 32038, 950;
1, 1250, 43847, 229881, 408217, 201956, 16819;
...
Illustration for n = 5:
The free polyominoes with five cells are also called free pentominoes.
For k = 1 there is only one free pentomino of width 1 as shown below, so T(5,1) = 1.
_
|_|
|_|
|_|
|_|
|_|
.
For k = 2 there are five free pentominoes of width 2 as shown below, so T(5,2) = 5.
_ _ _
|_| _|_| _|_| _ _ _ _
|_| |_|_| |_|_| |_|_| |_|_|
|_|_ |_| |_| |_|_| |_|_
|_|_| |_| |_| |_| |_|_|
.
For k = 3 there are six free pentominoes of width 3 as shown below, so T(5,3) = 6.
_ _ _ _ _ _ _ _ _ _
_|_|_| |_|_|_| |_| |_|_ _|_|_ |_|_|
|_|_| |_| |_|_ _ |_|_|_ |_|_|_| |_|_
|_| |_| |_|_|_| |_|_| |_| |_|_|
.
Therefore the 5th row of the triangle is [1, 5, 6] and the row sum is A000105(5) = 12.
.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Jan 07 2025
EXTENSIONS
a(21)-a(56) from Pontus von Brömssen, Jan 11 2025
STATUS
approved