This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379625 #53 Feb 17 2025 22:37:14 %S A379625 1,0,1,1,0,1,1,3,0,1,6,2,3,0,1,7,16,6,5,0,1,25,39,27,11,5,0,1,80,120, %T A379625 97,45,19,7,0,1,255,425,307,191,71,28,7,0,1,795,1565,1077,706,347,115, %U A379625 40,9,0,1,2919,5217,4170,2505,1454,574,171,53,9,0,1,10378,18511,15164,10069,5481,2740,919,257,69,11,0,1 %N A379625 Triangle read by rows: T(n,k) is the number of free polyominoes with n cells whose difference between length and width is k, n >= 1, k >= 0. %C A379625 Here the length is the longer of the two dimensions and the width is the shorter of the two dimensions. %H A379625 John Mason, <a href="/A379625/b379625.txt">Table of n, a(n) for n = 1..171</a> (first 18 rows) %H A379625 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>. %e A379625 Triangle begins: %e A379625 1; %e A379625 0, 1; %e A379625 1, 0, 1; %e A379625 1, 3, 0, 1; %e A379625 6, 2, 3, 0, 1; %e A379625 7, 16, 6, 5, 0, 1; %e A379625 25, 39, 27, 11, 5, 0, 1; %e A379625 80, 120, 97, 45, 19, 7, 0, 1; %e A379625 255, 425, 307, 191, 71, 28, 7, 0, 1; %e A379625 795, 1565, 1077, 706, 347, 115, 40, 9, 0, 1; %e A379625 2919, 5217, 4170, 2505, 1454, 574, 171, 53, 9, 0, 1; %e A379625 10378, 18511, 15164, 10069, 5481, 2740, 919, 257, 69, 11, 0, 1; %e A379625 ... %e A379625 Illustration for n = 5: %e A379625 The free polyominoes with five cells are also called free pentominoes. %e A379625 For k = 0 there are six free pentominoes with length 3 and width 3 as shown below, thus the difference between length and width is 3 - 3 = 0, so T(5,0) = 6. %e A379625 _ _ _ _ _ _ _ _ _ _ %e A379625 _|_|_| |_|_|_| |_| |_|_ _|_|_ |_|_| %e A379625 |_|_| |_| |_|_ _ |_|_|_ |_|_|_| |_|_ %e A379625 |_| |_| |_|_|_| |_|_| |_| |_|_| %e A379625 . %e A379625 For k = 1 there are two free pentominoes with length 3 and width 2 as shown below, thus the difference between length and width is 3 - 2 = 1, so T(5,1) = 2. %e A379625 _ _ _ _ %e A379625 |_|_| |_|_| %e A379625 |_|_| |_|_ %e A379625 |_| |_|_| %e A379625 . %e A379625 For k = 2 there are three free pentominoes with length 4 and width 2 as shown below, thus the difference between length and width is 4 - 2 = 2, so T(5,2) = 3. %e A379625 _ _ _ %e A379625 |_| _|_| _|_| %e A379625 |_| |_|_| |_|_| %e A379625 |_|_ |_| |_| %e A379625 |_|_| |_| |_| %e A379625 . %e A379625 For k = 3 there are no free pentominoes whose difference between length and width is 3, so T(5,3) = 0. %e A379625 For k = 4 there is only one free pentomino with length 5 and width 1 as shown below, thus the difference between length and width is 5 - 1 = 4, so T(5,4) = 1. %e A379625 _ %e A379625 |_| %e A379625 |_| %e A379625 |_| %e A379625 |_| %e A379625 |_| %e A379625 . %e A379625 Therefore the 5th row of the triangle is [6, 2, 3, 0, 1] and the row sum is A000105(5) = 12. %e A379625 Note that for n = 6 and k = 1 there are 15 free polyominoes with length 4 and width 3 thus the difference between length and width is 4 - 3 = 1. Also there is a free polyomino with length 3 and width 2 thus the difference between length and width is 3 - 2 = 1, so T(6,1) = 15 + 1 = 16. %e A379625 . %Y A379625 Row sums give A000105. %Y A379625 Column 1 gives A259088. %Y A379625 Row sums except the column 1 give A259087. %Y A379625 Leading diagonal gives A000012. %Y A379625 Second diagonal gives A000004. %Y A379625 Cf. A109613, A379623, A379624, A379626, A379627, A379628, A379629, A379637, A379638, A380283, A380284. %K A379625 nonn,tabl %O A379625 1,8 %A A379625 _Omar E. Pol_, Jan 12 2025 %E A379625 Terms a(29) and beyond from _Jinyuan Wang_, Jan 13 2025