cp's OEIS Frontend

A365860 For every cell of a polyomino let c be the number of cells that are in the same row or in the same column (including itself). a(n) is the sum of the c values of all cells of all free polyominoes with n cells.

%I A365860 #75 Apr 03 2024 15:26:51
%S A365860 1,4,16,62,206,790,3042,12648,52181,220372,927333,3917738,16491489,
%T A365860 69356256,290882884,1217780926
%N A365860 For every cell of a polyomino let c be the number of cells that are in the same row or in the same column (including itself). a(n) is the sum of the c values of all cells of all free polyominoes with n cells.
%C A365860 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.
%C A365860 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.
%C A365860 For another version and further information see A365835, which first differs at a(5).
%H A365860 Rodolfo Kurchan, <a href="https://www.puzzlefun.online/problems">Puzzle Fun, Problems, Colored Polyominoes</a>
%F A365860 a(n) == A057766(n) (mod 2). - _Pontus von Brömssen_, Sep 21 2023
%e A365860 For n = 1 the monomino has only one cell, so a(1) = 1.
%e A365860 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.
%e A365860 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.
%e A365860 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.
%e A365860 Three examples from the twelve pentominoes:
%e A365860 The I-pentomino with its c values looks like this:
%e A365860       +---+
%e A365860       | 5 |
%e A365860       +---+
%e A365860       | 5 |
%e A365860       +---+
%e A365860       | 5 |
%e A365860       +---+
%e A365860       | 5 |
%e A365860       +---+
%e A365860       | 5 |
%e A365860       +---+
%e A365860 The sum of the c values is 5 + 5 + 5 + 5 + 5 = 5^2 = 25, the maximum possible.
%e A365860 .
%e A365860 The U-pentomino with its c values looks like this:
%e A365860   +---+   +---+
%e A365860   | 3 |   | 3 |
%e A365860   +---+---+---+
%e A365860   | 4 | 3 | 4 |
%e A365860   +---+---+---+
%e A365860 The sum of the c values is 4 + 4 + 3 + 3 + 3 = 17.
%e A365860 .
%e A365860 The W-pentomino with its c values looks like this:
%e A365860   +---+
%e A365860   | 2 |
%e A365860   +---+---+
%e A365860   | 3 | 3 |
%e A365860   +---+---+---+
%e A365860       | 3 | 2 |
%e A365860       +---+---+
%e A365860 The sum of the c values is 3 + 3 + 3 + 2 + 2 = 3*(5-2) + 4 = 13, the minimum possible.
%e A365860 .
%Y A365860 Cf. A000105, A000290, A002378, A016777, A057766, A279019, A365835.
%K A365860 nonn,more
%O A365860 1,2
%A A365860 _Rodolfo Kurchan_ and _Omar E. Pol_, Sep 20 2023
%E A365860 a(7)-a(9) from _George Sicherman_, Sep 20 2023
%E A365860 a(10)-a(13) from _Pontus von Brömssen_, Sep 21 2023
%E A365860 a(14)-a(16) from _Pontus von Brömssen_, Apr 03 2024