OFFSET
1,2
COMMENTS
For a polyomino with n cells the maximum sum possible of the c values equals n^2 = A000290(n) and the minumum sum possible of the c values equals 3*(n - 2) + 4 = A016777(n-1). Hence the difference between the maximum possible and the minimum possible sum of the c 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.
Note that the concept "c 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 and further information see A365835, which first differs at a(5).
LINKS
Rodolfo Kurchan, Puzzle Fun, Problems, Colored Polyominoes
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 c values is 2 + 2 = 4, so a(2) = 4.
For n = 3 the sum of the c values of the I-tromino is 3 + 3 + 3 = 9 and the sum of the c values of the L-tromino is 3 + 2 + 2 = 7. The total sum of the c values is 9 + 7 = 16, so a(3) = 16.
For n = 4 the c values of the five (I, L, O, T, S) tetrominoes are 16, 12, 12, 12, 10 so the total sum of the c values is a(4) = 62.
Three examples from the twelve pentominoes:
The I-pentomino with its c values looks like this:
+---+
| 5 |
+---+
| 5 |
+---+
| 5 |
+---+
| 5 |
+---+
| 5 |
+---+
The sum of the c values is 5 + 5 + 5 + 5 + 5 = 5^2 = 25, the maximum possible.
.
The U-pentomino with its c values looks like this:
+---+ +---+
| 3 | | 3 |
+---+---+---+
| 4 | 3 | 4 |
+---+---+---+
The sum of the c values is 4 + 4 + 3 + 3 + 3 = 17.
.
The W-pentomino with its c values looks like this:
+---+
| 2 |
+---+---+
| 3 | 3 |
+---+---+---+
| 3 | 2 |
+---+---+
The sum of the c values is 3 + 3 + 3 + 2 + 2 = 3*(5-2) + 4 = 13, the minimum possible.
.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Rodolfo Kurchan and Omar E. Pol, Sep 20 2023
EXTENSIONS
a(7)-a(9) from George Sicherman, Sep 20 2023
a(10)-a(13) from Pontus von Brömssen, Sep 21 2023
a(14)-a(16) from Pontus von Brömssen, Apr 03 2024
STATUS
approved