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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.
2
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
1,2
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
LINKS
Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (first 10 rows)
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).
EXAMPLE
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.
CROSSREFS
Row lengths give A000105, n >= 1.
Right border gives A016777.
Row sums give A365835.
Sequence in context: A123157 A154684 A353071 * A256174 A096982 A212877
KEYWORD
nonn,tabf
AUTHOR
Rodolfo Kurchan and Omar E. Pol, Sep 22 2023
EXTENSIONS
Terms a(61) and beyond from Pontus von Brömssen, Oct 15 2023
STATUS
approved