A365835 For each cell of a polyomino let b be the number of cells that are in the same row or in the same column (including itself). Cells beyond gaps do not count. a(n) is the sum of the b values of all cells of all free polyominoes with n cells.
1, 4, 16, 62, 204, 776, 2936, 12030, 48783, 202734, 839239, 3489810, 14462593, 59906626, 247553908, 1021545890
Offset: 1
Examples
For n = 1 the monomino has only one cell, so a(1) = 1.
For n = 2 the domino has two cells. Each cell sees the other cell. The sum of the b values is 2 + 2 = 4, so a(2) = 4.
For n = 3 the sum of the b values of the I-tromino is 3 + 3 + 3 = 9 and the sum of the b values of the L-tromino is 3 + 2 + 2 = 7. The total sum is 9 + 7 = 16, so a(3) = 16.
For n = 4 the b values of the five tetrominoes (I, L, O, T, S) are 16, 12, 12, 12, 10, so the total sum of the b values is a(4) = 62.
Three examples from the twelve pentominoes:
The I-pentomino with its b values looks like this:
+---+
| 5 |
+---+
| 5 |
+---+
| 5 |
+---+
| 5 |
+---+
| 5 |
+---+
The sum of the b values is 5 + 5 + 5 + 5 + 5 = 5^2 = A000290(5) = 25, the maximum possible.
.
The U-pentomino with its b values looks like this:
+---+ +---+
| 2 | | 2 |
+---+---+---+
| 4 | 3 | 4 |
+---+---+---+
The sum of the b values is 4 + 4 + 3 + 2 + 2 = 15.
.
The W-pentomino with its b values looks like this:
+---+
| 2 |
+---+---+
| 3 | 3 |
+---+---+---+
| 3 | 2 |
+---+---+
The sum of the b values is 3 + 3 + 3 + 2 + 2 = 3*(5-2) + 4 = A016777(5-1) = 13, the minimum possible.
.
Links
- Rodolfo Kurchan, Puzzle Fun, Problems, Colored Polyominoes.
- George Sicherman, A colored version of the free pentominoes.
- Wikipedia, Polyomino.
- Wikipedia, Art gallery problem.
Formula
a(n) == A057766(n) (mod 2). - Pontus von Brömssen, Sep 21 2023
Extensions
a(6)-a(9) from George Sicherman, Sep 20 2023
a(6)-a(9) corrected and a(10)-a(13) added by Pontus von Brömssen, Sep 21 2023
a(14)-a(16) from Pontus von Brömssen, Apr 03 2024
Comments