A380287 Sum of the perimeters of the free polyominoes with n cells.

4, 6, 16, 48, 142, 472, 1670, 6364, 24604, 97668, 390070, 1570560, 6334644, 25617062, 103669288, 419930444, 1701635046, 6898183050

Offset

1,1

Comments

The perimeters of any holes are included here.

Links

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

Index entries for sequences related to polyominoes.

Formula

a(n) = Sum_{k=2..n+1} 2*k*A342243(n,k). - Pontus von Brömssen, Jan 27 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:
    _
   |_|     _       _       _
   |_|    |_|     |_|_    |_|_     _ _
   |_|    |_|_    |_|_|   |_|_|   |_|_|
   |_|    |_|_|     |_|   |_|     |_|_|
.
From left to right the perimeters are respectively [10, 10, 10, 10, 8] as shown below:
    _
   | |     _       _       _
   | |    | |     | |_    | |_     _ _
   | |    | |_    |_  |   |  _|   |   |
   |_|    |_ _|     |_|   |_|     |_ _|
.
The sum of the perimeters is 10 + 10 + 10 + 10 + 8 = 48, so a(4) = 48.
.

Crossrefs

Cf. A000105, A027709, A057730, A130622, A131482, A135942, A342243.

See A380575 for another version.

Sequence in context: A072279 A038236 A223269 * A264471 A264477 A223357

Adjacent sequences: A380284 A380285 A380286 * A380288 A380289 A380290

Keyword

nonn,more,changed

Author

Omar E. Pol, Jan 25 2025

Extensions

a(6)-a(18) (using A342243 b-file) from Pontus von Brömssen, Jan 27 2025

Status

approved