A153490 Sierpinski carpet, read by antidiagonals.

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, 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, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1

Offset

1,1

Comments

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.

Row sums are {1, 2, 2, 4, 5, 4, 6, 6, 4, 8, 10, 8, ...}.

Links

Table of n, a(n) for n=1..78.

Eric Weisstein's World of Mathematics, Sierpinski Carpet.

Wikipedia, Sierpinski carpet.

Example

The Sierpinski carpet matrix reads
   1 1 1 1 1 1 1 1 1 ...
   1 0 1 1 0 1 1 0 1 ...
   1 1 1 1 1 1 1 1 1 ...
   1 1 1 0 0 0 1 1 1 ...
   1 0 1 0 0 0 1 0 1 ...
   1 1 1 0 0 0 1 1 1 ...
   1 1 1 1 1 1 1 1 1 ...
   1 0 1 1 0 1 1 0 1 ...
   1 1 1 1 1 1 1 1 1 ...
   (...)
so the antidiagonals are
  {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, 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, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1},
  ...

Mathematica

<< MathWorld`Fractal`; fractal = SierpinskiCarpet;
a = fractal[4]; Table[Table[a[[m]][[n - m + 1]], {m, 1, n}], {n, 1, 12}];
Flatten[%]

Prog

(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

Crossrefs

Cf. A292688 (n-th antidiagonal concatenated as binary number), A292689 (decimal representation of these binary numbers).

Cf. A293143 (number of vertex points in a Sierpinski Carpet).

Sequence in context: A333922 A071026 A259022 * A014194 A014379 A014164

Adjacent sequences: A153487 A153488 A153489 * A153491 A153492 A153493

Keyword

nonn,tabl

Author

Roger L. Bagula, Dec 27 2008

Extensions

Edited by M. F. Hasler, Oct 20 2017

Status

approved