A379628 Total area between the free polyominoes with n cells and their bounding boxes.

0, 0, 1, 6, 35, 143, 631, 2747, 12027, 52470, 227993, 985522, 4235295, 18114067, 77112058, 327000797, 1381807943, 5821692946

Offset: 1

Author

Omar E. Pol, Jan 07 2025

Keywords

• nonn
• more

Comments

a(n) includes the area of 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:
    _
   | |     _ _     _ _     _ _
   | |    |   |   |   |   |   |    _ _
   | |    |   |   |   |   |   |   |   |
   |_|    |_ _|   |_ _|   |_ _|   |_ _|
.
  4 x 1   3 X 2   3 X 2   3 X 2   2 X 2
.
The total area of the free tetrominoes is 5*4 = 20.
The total area of the bounding boxes is 4 + 6 + 6 + 6 + 4 = 26.
The total area between the bounding boxes and the free tetrominoes is 26 - 20 = 6, so a(4) = 6.
.
		

Links

• Index entries for sequences related to polyominoes.

Crossrefs

Cf. A000105, A057766, A379623, A379624, A379627.

Formula

a(n) = A379627(n) - A057766(n).

Extensions

a(7)-a(16) from Pontus von Brömssen, Jan 11 2025

a(17)-a(18) from John Mason, Feb 16 2025