A153490 - OEIS

%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