

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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 bvalues 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



FORMULA

For n >= 1; T(n,1) = n^2.
For n >= 3; T(n,2) = (n  1)^2 + 3 = A117950(n1).
For n >= 4; T(n,3) = (n  1)^2 + 3 = A117950(n1).


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 kth pentomino from the diagram.
For further information see also A365835.


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



EXTENSIONS



STATUS

approved



