A365835 -id:A365835 - cp's OEIS frontend

A365906 Irregular triangle T(n,k) read by rows, n>=1, k>=1, in which row n lists in nonincreasing order the sum of the b values (described in A365835) of the cells of every free polyomino with n cells.

1, 4, 9, 7, 16, 12, 12, 12, 10, 25, 19, 19, 17, 17, 17, 17, 15, 15, 15, 15, 13, 36, 28, 28, 28, 24, 24, 24, 24, 24, 24, 24, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 20, 20, 20, 20, 20, 20, 20, 18, 18, 18, 18, 18, 16, 49, 39, 39, 39, 33, 33, 33, 33, 33, 33

Offset: 1

Rodolfo Kurchan and Omar E. Pol, Sep 22 2023

Observation: at least for 1 <= n <= 6 the parity of the terms in row n coincides with the parity of n. If n is odd then every polyomino has an odd number of odd b values, otherwise if n is even then every polyomino has an even number of odd b values.

The preceding observation is true for all n, because the b-values count each cell once (it is in the same row/column as itself) and pairs of distinct cells in the same row or column (with no gaps in between) twice (once in each direction). - Pontus von Brömssen, Oct 15 2023

			Triangle begins:
   1;
   4;
   9,  7;
  16, 12, 12, 12, 10;
  25, 19, 19, 17, 17, 17, 17, 15, 15, 15, 15, 13;
  ...
For n = 5 the twelve pentominoes and the b values of their cells are as shown below:
.
      I     L         Y      P        T       V           X
.     _     _         _     _ _     _ _ _     _           _
     |_|   |_|      _|_|   |_|_|   |_|_|_|   |_|        _|_|_
     |_|   |_|     |_|_|   |_|_|     |_|     |_|_ _    |_|_|_|
     |_|   |_|_      |_|   |_|       |_|     |_|_|_|     |_|
     |_|   |_|_|     |_|
     |_|
      5     4         4     4 3     3 5 3     3           3
      5     4       2 5     4 3       3       3         3 5 3
      5     4         4     3         3       5 3 3       3
      5     5 2       4
      5
.
      F        N        U         Z         W
.     _ _       _     _   _     _ _       _
    _|_|_|    _|_|   |_|_|_|   |_|_|     |_|_
   |_|_|     |_|_|   |_|_|_|     |_|_    |_|_|_
     |_|     |_|                 |_|_|     |_|_|
             |_|
      4 2       2     2   2     2 4       2
    2 4       4 3     4 3 4       3       3 3
      3       3                   4 2       3 2
              3
.
T(5,k) is the sum of the b values of all cells of the k-th pentomino from the diagram.
For further information see also A365835.

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (first 10 rows)
Rodolfo Kurchan, Puzzle Fun, Problems, Colored Polyominoes
George Sicherman, A colored version of the free pentominoes

Row lengths give A000105, n >= 1.
Right border gives A016777.
Row sums give A365835.
Cf. A000035, A000290, A002378, A057766, A117950, A279019, A365860.

For n >= 1; T(n,1) = n^2.
For n >= 3; T(n,2) = (n - 1)^2 + 3 = A117950(n-1).
For n >= 4; T(n,3) = (n - 1)^2 + 3 = A117950(n-1).

Terms a(61) and beyond from Pontus von Brömssen, Oct 15 2023

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.

Original entry on oeis.org

1, 4, 16, 62, 206, 790, 3042, 12648, 52181, 220372, 927333, 3917738, 16491489, 69356256, 290882884, 1217780926

Offset: 1

Rodolfo Kurchan and Omar E. Pol, Sep 20 2023

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).

			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.
.

Rodolfo Kurchan, Puzzle Fun, Problems, Colored Polyominoes

Cf. A000105, A000290, A002378, A016777, A057766, A279019, A365835.

a(n) == A057766(n) (mod 2). - Pontus von Brömssen, Sep 21 2023

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

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A365906 Irregular triangle T(n,k) read by rows, n>=1, k>=1, in which row n lists in nonincreasing order the sum of the b values (described in A365835) of the cells of every free polyomino with n cells.

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