A379624 Triangle read by rows: T(n,k) is the number of free polyominoes with n cells and length k, n >= 1, k = 1..n.

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

Offset: 1

Author

Omar E. Pol, Jan 07 2025

Keywords

• nonn
• tabl

Comments

The length here is the longer of the two dimensions.

Examples

			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.
.
		

Links

• John Mason, Table of n, a(n) for n = 1..171 (first 18 rows)
• Index entries for sequences related to polyominoes.

Crossrefs

Row sums give A000105(n).

Column 1 gives A000007.

Leading diagonal gives A000012.

For free polyominoes of width k see A379623.

Cf. A109613, A379627, A379628.

Extensions

Terms a(37) and beyond from Jinyuan Wang, Jan 08 2025