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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.

%I A365906 #68 Oct 16 2023 03:34:06
%S A365906 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 A365906 28,24,24,24,24,24,24,24,22,22,22,22,22,22,22,22,22,22,22,20,20,20,20,
%U A365906 20,20,20,18,18,18,18,18,16,49,39,39,39,33,33,33,33,33,33
%N 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.
%C A365906 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 A365906 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 A365906 Pontus von Brömssen, <a href="/A365906/b365906.txt">Table of n, a(n) for n = 1..6473</a> (first 10 rows)
%H A365906 Rodolfo Kurchan, <a href="https://www.puzzlefun.online/problems">Puzzle Fun, Problems, Colored Polyominoes</a>
%H A365906 George Sicherman, <a href="http://static.wixstatic.com/media/799cc4_0fb1e118c87c4e84903509920c43f42e~mv2.png">A colored version of the free pentominoes</a>
%F A365906 For n >= 1; T(n,1) = n^2.
%F A365906 For n >= 3; T(n,2) = (n - 1)^2 + 3 = A117950(n-1).
%F A365906 For n >= 4; T(n,3) = (n - 1)^2 + 3 = A117950(n-1).
%e A365906 Triangle begins:
%e A365906    1;
%e A365906    4;
%e A365906    9,  7;
%e A365906   16, 12, 12, 12, 10;
%e A365906   25, 19, 19, 17, 17, 17, 17, 15, 15, 15, 15, 13;
%e A365906   ...
%e A365906 For n = 5 the twelve pentominoes and the b values of their cells are as shown below:
%e A365906 .
%e A365906       I     L         Y      P        T       V           X
%e A365906 .     _     _         _     _ _     _ _ _     _           _
%e A365906      |_|   |_|      _|_|   |_|_|   |_|_|_|   |_|        _|_|_
%e A365906      |_|   |_|     |_|_|   |_|_|     |_|     |_|_ _    |_|_|_|
%e A365906      |_|   |_|_      |_|   |_|       |_|     |_|_|_|     |_|
%e A365906      |_|   |_|_|     |_|
%e A365906      |_|
%e A365906       5     4         4     4 3     3 5 3     3           3
%e A365906       5     4       2 5     4 3       3       3         3 5 3
%e A365906       5     4         4     3         3       5 3 3       3
%e A365906       5     5 2       4
%e A365906       5
%e A365906 .
%e A365906       F        N        U         Z         W
%e A365906 .     _ _       _     _   _     _ _       _
%e A365906     _|_|_|    _|_|   |_|_|_|   |_|_|     |_|_
%e A365906    |_|_|     |_|_|   |_|_|_|     |_|_    |_|_|_
%e A365906      |_|     |_|                 |_|_|     |_|_|
%e A365906              |_|
%e A365906       4 2       2     2   2     2 4       2
%e A365906     2 4       4 3     4 3 4       3       3 3
%e A365906       3       3                   4 2       3 2
%e A365906               3
%e A365906 .
%e A365906 T(5,k) is the sum of the b values of all cells of the k-th pentomino from the diagram.
%e A365906 For further information see also A365835.
%Y A365906 Row lengths give A000105, n >= 1.
%Y A365906 Right border gives A016777.
%Y A365906 Row sums give A365835.
%Y A365906 Cf. A000035, A000290, A002378, A057766, A117950, A279019, A365860.
%K A365906 nonn,tabf
%O A365906 1,2
%A A365906 _Rodolfo Kurchan_ and _Omar E. Pol_, Sep 22 2023
%E A365906 Terms a(61) and beyond from _Pontus von Brömssen_, Oct 15 2023