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%I #18 Apr 29 2018 02:12:11
%S 1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,1,1,1,1,0,1,1,1,1,1,1,0,0,1,
%T 1,1,1,0,1,0,0,0,1,0,1,1,1,1,1,0,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,0,
%U 1,1,0,1,1,0,1,1,0,1
%N Sierpinski carpet, read by antidiagonals.
%C The Sierpinski carpet is the fractal obtained by starting with a unit square and at subsequent iterations, subdividing each square into 3 X 3 smaller squares and removing (from nonempty squares) the middle square. After the n-th iteration, the upper-left 3^n X 3^n squares will always remain the same. Therefore this sequence, which reads these by antidiagonals, is well-defined.
%C Row sums are {1, 2, 2, 4, 5, 4, 6, 6, 4, 8, 10, 8, ...}.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiCarpet.html">Sierpinski Carpet</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sierpinski_carpet">Sierpinski carpet</a>.
%e The Sierpinski carpet matrix reads
%e 1 1 1 1 1 1 1 1 1 ...
%e 1 0 1 1 0 1 1 0 1 ...
%e 1 1 1 1 1 1 1 1 1 ...
%e 1 1 1 0 0 0 1 1 1 ...
%e 1 0 1 0 0 0 1 0 1 ...
%e 1 1 1 0 0 0 1 1 1 ...
%e 1 1 1 1 1 1 1 1 1 ...
%e 1 0 1 1 0 1 1 0 1 ...
%e 1 1 1 1 1 1 1 1 1 ...
%e (...)
%e so the antidiagonals are
%e {1},
%e {1, 1},
%e {1, 0, 1},
%e {1, 1, 1, 1},
%e {1, 1, 1, 1, 1},
%e {1, 0, 1, 1, 0, 1},
%e {1, 1, 1, 0, 1, 1, 1},
%e {1, 1, 1, 0, 0, 1, 1, 1},
%e {1, 0, 1, 0, 0, 0, 1, 0, 1},
%e {1, 1, 1, 1, 0, 0, 1, 1, 1, 1},
%e {1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1},
%e {1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1},
%e ...
%t << MathWorld`Fractal`; fractal = SierpinskiCarpet;
%t a = fractal[4]; Table[Table[a[[m]][[n - m + 1]], {m, 1, n}], {n, 1, 12}];
%t Flatten[%]
%o (PARI) A153490_row(n,A=Mat(1))={while(#A<n,A=matrix(3*#A,3*#A,i,j,if(A[(i+2)\3,(j+2)\3],i%3!=2||j%3!=2)));vector(n,k,A[k,n-k+1])} \\ _M. F. Hasler_, Oct 23 2017
%Y Cf. A292688 (n-th antidiagonal concatenated as binary number), A292689 (decimal representation of these binary numbers).
%Y Cf. A293143 (number of vertex points in a Sierpinski Carpet).
%K nonn,tabl
%O 1,1
%A _Roger L. Bagula_, Dec 27 2008
%E Edited by _M. F. Hasler_, Oct 20 2017
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