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 A225803 #29 Sep 06 2021 04:27:24 %S A225803 1,1,1,1,1,1,2,1,1,2,2,0,1,1,1,2,2,1,2,4,0,2,1,1,4,13,10,6,3,1,0,0,1, %T A225803 1,1,3,4,1,1,3,8,3,2,3,0,0,1,1,6,23,33,24,15,6,0,2,2,2,1,1,6,40,101, %U A225803 129,79,74,53,13,9,11,4,0,0,0,0,1 %N A225803 Number T(n,k,u) of tilings of an n X k rectangle using integer-sided square tiles, reduced for symmetry, containing u nodes that are unconnected to any of their neighbors; irregular triangle T(n,k,u), 1 <= k < n, u >= 0, read by rows. %C A225803 The number of entries per row is given by A225568(n>0 and n != A000217(1:)). %H A225803 Christopher Hunt Gribble, <a href="/A225803/b225803.txt">Rows 1..28 for n = 2..8 and k = 1..n-1 flattened</a> %H A225803 Christopher Hunt Gribble, <a href="/A225803/a225803.cpp.txt">C++ program</a> %F A225803 T1(n,k,0) = 1, T1(n,k,1) = floor(n/2)*floor(k/2). %e A225803 The irregular triangle T(n,k,u) begins: %e A225803 n,k\u 0 1 2 3 4 5 6 7 8 9 10 11 12 ... %e A225803 2,1 1 %e A225803 3,1 1 %e A225803 3,2 1 1 %e A225803 4,1 1 %e A225803 4,2 1 2 1 %e A225803 4,3 1 2 2 0 1 %e A225803 5,1 1 %e A225803 5,2 1 2 2 %e A225803 5,3 1 2 4 0 2 1 %e A225803 5,4 1 4 13 10 6 3 1 0 0 1 %e A225803 6,1 1 %e A225803 6,2 1 3 4 1 %e A225803 6,3 1 3 8 3 2 3 0 0 1 %e A225803 6,4 1 6 23 33 24 15 6 0 2 2 1 %e A225803 6,5 1 6 40 101 79 74 53 13 9 11 4 0 0 ... %e A225803 ... %e A225803 T(5,3,2) = 4 because there are 4 different sets of tilings of the 5 X 3 rectangle by integer-sided squares in which each tiling contains 2 isolated nodes. Any sequence of group D2 operations will transform each tiling in a set into another in the same set. Group D2 operations are: %e A225803 . the identity operation %e A225803 . rotation by 180 degrees %e A225803 . reflection about a horizontal axis through the center %e A225803 . reflection about a vertical axis through the center %e A225803 A 2 X 2 square contains 1 isolated node. Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors. An example of a tiling in each set is: %e A225803 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 %e A225803 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 %e A225803 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 %e A225803 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 %e A225803 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 %e A225803 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 %Y A225803 Cf. A224239, A227004, A227690, A224850, A224861, A224867, A225777, A225542. %K A225803 nonn,tabf %O A225803 1,7 %A A225803 _Christopher Hunt Gribble_, Jul 28 2013