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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 (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 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) Rodolfo Kurchan, Puzzle Fun, Problems, Colored Polyominoes George Sicherman, A colored version of the free pentominoes 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. Cf. A000035, A000290, A002378, A057766, A117950, A279019, A365860. Sequence in context: A123157 A154684 A353071 * A256174 A096982 A212877 Adjacent sequences: A365903 A365904 A365905 * A365907 A365908 A365909 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

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Last modified August 10 02:46 EDT 2024. Contains 375044 sequences. (Running on oeis4.)