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

1,4

Comments

The regions include any holes in the polyominoes.

Links

Table of n, a(n) for n=1..18.

Index entries for sequences related to polyominoes.

Formula

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

Example

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.
.

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.

Sequence in context: A271492 A202462 A354701 * A055280 A268304 A273450

Adjacent sequences: A380282 A380283 A380284 * A380286 A380287 A380288

Keyword

nonn,more

Author

Omar E. Pol, Jan 18 2025

Extensions

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

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

Status

approved