login
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

%I #68 Oct 16 2023 03:34:06

%S 1,4,9,7,16,12,12,12,10,25,19,19,17,17,17,17,15,15,15,15,13,36,28,28,

%T 28,24,24,24,24,24,24,24,22,22,22,22,22,22,22,22,22,22,22,20,20,20,20,

%U 20,20,20,18,18,18,18,18,16,49,39,39,39,33,33,33,33,33,33

%N 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.

%C 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.

%C 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

%H Pontus von Brömssen, <a href="/A365906/b365906.txt">Table of n, a(n) for n = 1..6473</a> (first 10 rows)

%H Rodolfo Kurchan, <a href="https://www.puzzlefun.online/problems">Puzzle Fun, Problems, Colored Polyominoes</a>

%H George Sicherman, <a href="http://static.wixstatic.com/media/799cc4_0fb1e118c87c4e84903509920c43f42e~mv2.png">A colored version of the free pentominoes</a>

%F For n >= 1; T(n,1) = n^2.

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

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

%e Triangle begins:

%e 1;

%e 4;

%e 9, 7;

%e 16, 12, 12, 12, 10;

%e 25, 19, 19, 17, 17, 17, 17, 15, 15, 15, 15, 13;

%e ...

%e For n = 5 the twelve pentominoes and the b values of their cells are as shown below:

%e .

%e I L Y P T V X

%e . _ _ _ _ _ _ _ _ _ _

%e |_| |_| _|_| |_|_| |_|_|_| |_| _|_|_

%e |_| |_| |_|_| |_|_| |_| |_|_ _ |_|_|_|

%e |_| |_|_ |_| |_| |_| |_|_|_| |_|

%e |_| |_|_| |_|

%e |_|

%e 5 4 4 4 3 3 5 3 3 3

%e 5 4 2 5 4 3 3 3 3 5 3

%e 5 4 4 3 3 5 3 3 3

%e 5 5 2 4

%e 5

%e .

%e F N U Z W

%e . _ _ _ _ _ _ _ _

%e _|_|_| _|_| |_|_|_| |_|_| |_|_

%e |_|_| |_|_| |_|_|_| |_|_ |_|_|_

%e |_| |_| |_|_| |_|_|

%e |_|

%e 4 2 2 2 2 2 4 2

%e 2 4 4 3 4 3 4 3 3 3

%e 3 3 4 2 3 2

%e 3

%e .

%e T(5,k) is the sum of the b values of all cells of the k-th pentomino from the diagram.

%e For further information see also A365835.

%Y Row lengths give A000105, n >= 1.

%Y Right border gives A016777.

%Y Row sums give A365835.

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

%K nonn,tabf

%O 1,2

%A _Rodolfo Kurchan_ and _Omar E. Pol_, Sep 22 2023

%E Terms a(61) and beyond from _Pontus von Brömssen_, Oct 15 2023