cp's OEIS Frontend

A153490 Sierpinski carpet, read by antidiagonals.

%I A153490 #19 Feb 16 2025 08:33:09
%S A153490 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 A153490 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 A153490 1,1,0,1,1,0,1,1,0,1
%N A153490 Sierpinski carpet, read by antidiagonals.
%C A153490 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 A153490 Row sums are {1, 2, 2, 4, 5, 4, 6, 6, 4, 8, 10, 8, ...}.
%H A153490 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiCarpet.html">Sierpinski Carpet</a>.
%H A153490 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sierpinski_carpet">Sierpinski carpet</a>.
%e A153490 The Sierpinski carpet matrix reads
%e A153490    1 1 1 1 1 1 1 1 1 ...
%e A153490    1 0 1 1 0 1 1 0 1 ...
%e A153490    1 1 1 1 1 1 1 1 1 ...
%e A153490    1 1 1 0 0 0 1 1 1 ...
%e A153490    1 0 1 0 0 0 1 0 1 ...
%e A153490    1 1 1 0 0 0 1 1 1 ...
%e A153490    1 1 1 1 1 1 1 1 1 ...
%e A153490    1 0 1 1 0 1 1 0 1 ...
%e A153490    1 1 1 1 1 1 1 1 1 ...
%e A153490    (...)
%e A153490 so the antidiagonals are
%e A153490   {1},
%e A153490   {1, 1},
%e A153490   {1, 0, 1},
%e A153490   {1, 1, 1, 1},
%e A153490   {1, 1, 1, 1, 1},
%e A153490   {1, 0, 1, 1, 0, 1},
%e A153490   {1, 1, 1, 0, 1, 1, 1},
%e A153490   {1, 1, 1, 0, 0, 1, 1, 1},
%e A153490   {1, 0, 1, 0, 0, 0, 1, 0, 1},
%e A153490   {1, 1, 1, 1, 0, 0, 1, 1, 1, 1},
%e A153490   {1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1},
%e A153490   {1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1},
%e A153490   ...
%t A153490 << MathWorld`Fractal`; fractal = SierpinskiCarpet;
%t A153490 a = fractal[4]; Table[Table[a[[m]][[n - m + 1]], {m, 1, n}], {n, 1, 12}];
%t A153490 Flatten[%]
%o A153490 (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 A153490 Cf. A292688 (n-th antidiagonal concatenated as binary number), A292689 (decimal representation of these binary numbers).
%Y A153490 Cf. A293143 (number of vertex points in a Sierpinski Carpet).
%K A153490 nonn,tabl
%O A153490 1,1
%A A153490 _Roger L. Bagula_, Dec 27 2008
%E A153490 Edited by _M. F. Hasler_, Oct 20 2017