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A379623 Irregular triangle read by rows: T(n,k) is the number of free polyominoes with n cells and width k, n >= 1, 1 <= k <= ceiling(n/2).

%I A379623 #47 Feb 17 2025 22:37:09
%S A379623 1,1,1,1,1,4,1,5,6,1,12,22,1,18,71,18,1,37,193,138,1,60,490,661,73,1,
%T A379623 117,1221,2547,769,1,200,3011,8417,5189,255,1,379,7393,26164,25920,
%U A379623 3743,1,669,18025,78074,108834,32038,950,1,1250,43847,229881,408217,201956,16819
%N A379623 Irregular triangle read by rows: T(n,k) is the number of free polyominoes with n cells and width k, n >= 1, 1 <= k <= ceiling(n/2).
%C A379623 The width here is the shorter of the two dimensions.
%H A379623 John Mason, <a href="/A379623/b379623.txt">Table of n, a(n) for n = 1..90</a> (first 18 rows)
%H A379623 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%e A379623 Triangle begins:
%e A379623   1;
%e A379623   1;
%e A379623   1,    1;
%e A379623   1,    4;
%e A379623   1,    5,     6;
%e A379623   1,   12,    22;
%e A379623   1,   18,    71,     18;
%e A379623   1,   37,   193,    138;
%e A379623   1,   60,   490,    661,     73;
%e A379623   1,  117,  1221,   2547,    769;
%e A379623   1,  200,  3011,   8417,   5189,    255;
%e A379623   1,  379,  7393,  26164,  25920,   3743;
%e A379623   1,  669, 18025,  78074, 108834,  32038,   950;
%e A379623   1, 1250, 43847, 229881, 408217, 201956, 16819;
%e A379623   ...
%e A379623 Illustration for n = 5:
%e A379623 The free polyominoes with five cells are also called free pentominoes.
%e A379623 For k = 1 there is only one free pentomino of width 1 as shown below, so T(5,1) = 1.
%e A379623    _
%e A379623   |_|
%e A379623   |_|
%e A379623   |_|
%e A379623   |_|
%e A379623   |_|
%e A379623 .
%e A379623 For k = 2 there are five free pentominoes of width 2 as shown below, so T(5,2) = 5.
%e A379623    _           _         _
%e A379623   |_|        _|_|      _|_|      _ _       _ _
%e A379623   |_|       |_|_|     |_|_|     |_|_|     |_|_|
%e A379623   |_|_      |_|         |_|     |_|_|     |_|_
%e A379623   |_|_|     |_|         |_|     |_|       |_|_|
%e A379623 .
%e A379623 For k = 3 there are six free pentominoes of width 3 as shown below, so T(5,3) = 6.
%e A379623      _ _     _ _ _     _         _           _       _ _
%e A379623    _|_|_|   |_|_|_|   |_|       |_|_       _|_|_    |_|_|
%e A379623   |_|_|       |_|     |_|_ _    |_|_|_    |_|_|_|     |_|_
%e A379623     |_|       |_|     |_|_|_|     |_|_|     |_|       |_|_|
%e A379623 .
%e A379623 Therefore the 5th row of the triangle is [1, 5, 6] and the row sum is A000105(5) = 12.
%e A379623 .
%Y A379623 Row sums give A000105(n).
%Y A379623 Row lengths give A110654(n).
%Y A379623 For free polyominoes of length k see A379624.
%Y A379623 Cf. A057051, A352720, A379627, A379628.
%K A379623 nonn,tabf
%O A379623 1,6
%A A379623 _Omar E. Pol_, Jan 07 2025
%E A379623 a(21)-a(56) from _Pontus von Brömssen_, Jan 11 2025