A379623 Irregular triangle read by rows: T(n,k) is the number of free polyominoes with n cells and width k, n >= 1, 1 <= k <= ceiling(n/2).

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

Offset: 1

Author

Omar E. Pol, Jan 07 2025

Keywords

• nonn
• tabf

Comments

The width here is the shorter of the two dimensions.

Examples

			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.
.
		

Links

• John Mason, Table of n, a(n) for n = 1..90 (first 18 rows)
• Index entries for sequences related to polyominoes.

Crossrefs

Row sums give A000105(n).

Row lengths give A110654(n).

For free polyominoes of length k see A379624.

Cf. A057051, A352720, A379627, A379628.

Extensions

a(21)-a(56) from Pontus von Brömssen, Jan 11 2025