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

Author

Rodolfo Kurchan and Omar E. Pol, Sep 22 2023

Keywords

• nonn
• tabf

Comments

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

Examples

			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.
		

Links

• 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

Crossrefs

Row lengths give A000105, n >= 1.

Right border gives A016777.

Row sums give A365835.

Cf. A000035, A000290, A002378, A057766, A117950, A279019, A365860.

Formula

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).

Extensions

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