OFFSET
1,2
COMMENTS
The sum of the b values of a polyomino seems to give an idea of its "alignment". It seems the highest values correspond to the most aligned polyominoes and the lowest values correspond to the least aligned polyominoes. For example, in an I-polyomino with n cells the sum of the b values equals n^2 = A000290(n), which is the maximum possible. Other polyominoes with k cells have a lower value.
A question from Jon E. Schoenfield: Is it true that the minimum sum of the b values (for a given value of n, the number of cells) is always obtained by only one polyomino, and that that is the one that can be built using a tight zigzag pattern (turning alternately to the left or right at each step) -- i.e., the monomino, the domino, the "L"-tromino, the "S"-tetromino, the "W"-pentomino, etc.?
The answer is: Yes. And the sum of the b values is equal to 3*(n - 2) + 4 = A016777(n-1), the minimum possible.
Hence the difference between the maximum possible and the minimum possible sum of the b values is A000290(n) - A016777(n-1) = A279019(n+3), n >= 1. Also it's equal to A002378(n-1) if n >= 2. See examples.
Resembles the art gallery problem.
Note that the concept "b value" for a cell or vertex can also be applied in other polyforms and in other types of graphs, for example: cellular automata, partitions, etc.
For another version see A365860, which first differs at a(5).
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
EXAMPLE
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.
.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Rodolfo Kurchan and Omar E. Pol, Sep 19 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
STATUS
approved