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

Author

Roger L. Bagula, Dec 27 2008

Keywords

• nonn
• tabl

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, ...}.

Examples

			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},
  ...
		

Links

• Eric Weisstein's World of Mathematics, Sierpinski Carpet.
• Wikipedia, Sierpinski carpet.

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).

Programs

Mathematica

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

PARI

A153490_row(n,A=Mat(1))={while(#AM. F. Hasler, Oct 23 2017

Extensions

Edited by M. F. Hasler, Oct 20 2017