A380285 Total number of regions between the free polyominoes with n cells and their bounding boxes.

0, 0, 1, 5, 21, 71, 255, 961, 3630, 13973, 53938, 209641, 815784, 3183642, 12439291, 48686549, 190787588, 748645732

Offset: 1

Author

Omar E. Pol, Jan 18 2025

Keywords

• nonn
• more

Comments

The regions include any holes in the polyominoes.

Examples

			Illustration for n = 4:
The free polyominoes with four cells are also called free tetrominoes.
The five free tetrominoes are as shown below:
    _
   |_|     _       _       _
   |_|    |_|     |_|_    |_|_     _ _
   |_|    |_|_    |_|_|   |_|_|   |_|_|
   |_|    |_|_|     |_|   |_|     |_|_|
.
The bounding boxes are respectively as shown below:
    _
   | |     _ _     _ _     _ _
   | |    |   |   |   |   |   |    _ _
   | |    |   |   |   |   |   |   |   |
   |_|    |_ _|   |_ _|   |_ _|   |_ _|
.
From left to right the number of regions between the free tetrominoes and their bounding boxes are respectively 0, 1, 2, 2, 0. Hence the total number of regions is 0 + 1 + 2 + 2 + 0 = 5, so a(4) = 5.
.
		

Links

• Index entries for sequences related to polyominoes.

Crossrefs

Row sums of A380283 and of A380284.

Cf. A379628 (total area of the regions).

Cf. A000105, A057766, A379623, A379624, A379625, A379626, A379627, A379629, A379637, A379638, A380282.

Formula

a(n) = Sum_{k>0} k*A380282(n,k). - Pontus von Brömssen, Jan 24 2025

Extensions

a(8)-a(16) from Pontus von Brömssen, Jan 24 2025

a(17)-a(18) from John Mason, Feb 14 2025