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A153490 Sierpinski carpet, read by antidiagonals. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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: A059095 A105597 A071026 * 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

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Last modified May 23 03:32 EDT 2018. Contains 304449 sequences. (Running on oeis4.)