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%I #62 Apr 06 2020 19:10:21
%S 1,1,2,1,3,1,2,9,7,4,4,5,2,25,11,40,8,33,3,16,0,4
%N 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 The sequence of row lengths is A261243. - _Wolfdieter Lang_, Aug 18 2015
%C 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 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 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 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Vicsek_fractal">Vicsek fractal</a>
%e Irregular triangle begins:
%e n\k 0 1 2 3 4 5 6 7 8 ...
%e 1 1
%e 2 1
%e 3 2 1
%e 4 3 1 2
%e 5 9 7 4 4 5 2
%e 6 25 11 40 8 33 3 16 0 4
%e ...
%Y Cf. A002276 (10 squares, 2 holes), A016203 (8 squares, 0 holes), A023001 (8 squares, 1 hole), A218724 (21 squares, 4 holes).
%Y Cf. A261242, A261243.
%K nonn,tabf,more
%O 1,3
%A _Kival Ngaokrajang_, Jun 06 2015
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