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 A258643 #62 Apr 06 2020 19:10:21 %S A258643 1,1,2,1,3,1,2,9,7,4,4,5,2,25,11,40,8,33,3,16,0,4 %N A258643 Irregular triangle read by rows, n >= 1, k >= 0: T(n,k) is the number of distinct patterns of n X n squares with k holes that are squares (see the construction rule in comments). %C A258643 The sequence of row lengths is A261243. - _Wolfdieter Lang_, Aug 18 2015 %C A258643 The construction rules are: (o) The n X n square has horizontal and vertical diagonals. (i) A pattern must be symmetric with respect to both vertical and horizontal axes. (ii) For n >= 2, each pattern must have four squares at the corners. (iii) The squares must have continuity contact to each other either by sides or corners. (iv) The hole(s) must be square(s). Mirror patterns with respect to the main diagonal are not considered as different. See illustration in the links. %C A258643 Each pattern can be a seed of a box fractal; e.g., the second pattern of T(3,0), consisting of 5 squares and 0 holes, is a seed of the Vicsek fractal (see a link below); the second pattern of T(4,2), consisting of 10 squares and 2 holes, is a seed of the fractal in a link of A002276. %C A258643 If the figures are rotated by 45 degrees in the clockwise direction they can be considered as binary bisymmetric n X n matrices B_n if a red square stand for 1 and an empty square for 0. The four corners have entries 1, that is B_n[1, 1] = 1 = B_n[1, n]. The continuity of the red squares, mentioned above in point (iii), means that there is no rectangular path of 0's (no diagonal steps) in the matrix B_n that dissects it into two parts. See A261242 for more details, where also the figures with nonsquare holes and the mirrors (row reversion in the B_n matrix) are considered. - _Wolfdieter Lang_, Aug 18 2015 %H A258643 Kival Ngaokrajang, <a href="/A258643/a258643_8.pdf">Illustration of pattern T(n,k) for n = 1..5, k >= 0</a>, <a href="/A258643/a258643_11.pdf">T(6,k) patterns</a> %H A258643 Wikipedia, <a href="http://en.wikipedia.org/wiki/Vicsek_fractal">Vicsek fractal</a> %e A258643 Irregular triangle begins: %e A258643 n\k 0 1 2 3 4 5 6 7 8 ... %e A258643 1 1 %e A258643 2 1 %e A258643 3 2 1 %e A258643 4 3 1 2 %e A258643 5 9 7 4 4 5 2 %e A258643 6 25 11 40 8 33 3 16 0 4 %e A258643 ... %Y A258643 Cf. A002276 (10 squares, 2 holes), A016203 (8 squares, 0 holes), A023001 (8 squares, 1 hole), A218724 (21 squares, 4 holes). %Y A258643 Cf. A261242, A261243. %K A258643 nonn,tabf,more %O A258643 1,3 %A A258643 _Kival Ngaokrajang_, Jun 06 2015