A292689 -id:A292689 - cp's OEIS frontend

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

Roger L. Bagula, Dec 27 2008

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, ...}.

			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},
  ...

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

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).

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

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

Edited by M. F. Hasler, Oct 20 2017

A292688 Antidiagonals of the Sierpinski carpet (as binary numbers).

Original entry on oeis.org

1, 11, 101, 1111, 11111, 101101, 1110111, 11100111, 101000101, 1111001111, 11111011111, 101101101101, 1111111111111, 11111111111111, 101101101101101, 1110111111110111, 11100111111100111, 101000101101000101, 1111001110111001111, 11111011100111011111, 101101101000101101101

Offset: 1

M. F. Hasler, Oct 23 2017

nonn

Concatenation of the terms in the rows of A153490.
The Sierpinski carpet A153490 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 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.
The n-th term a(n) has n digits. See A292689 for the decimal value of a(n) considered as binary number.
The Hamming weights (or sum of digits) of the terms (also row sums of A153490) are (1, 2, 2, 4, 5, 4, 6, 6, 4, 8, 10, 8, 13, 14, 10, 14, 13, 8, 14, 16, 12, 18, 18, 12, 16,...)

			The Sierpinski carpet matrix A153490 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...
   (...)
The concatenation of the terms in the antidiagonals yields 1, 11, 101, 1111, 11111, 101101, 1110111, 11100111, 101000101, 1111001111, 11111011111, 101101101101, 1111111111111, 11111111111111, 101101101101101, 1110111111110111, 11100111111100111, 101000101101000101, 1111001110111001111, ...

Paolo Xausa, Table of n, a(n) for n = 1..729
Eric Weisstein's World of Mathematics, Sierpinski Carpet.
Wikipedia, Sierpinski carpet.

Cf. A153490, A292689.

A292688[i_]:=With[{a=Nest[ArrayFlatten[{{#,#,#},{#,0,#},{#,#,#}}]&,{{1}},i]},Array[FromDigits[Diagonal[a,#]]&,3^i,1-3^i]];A292688[3] (* Paolo Xausa, May 13 2023 *)

PARI
```
A292688(n,A=Mat(1))={while(#A
				
```

A293974 Row sums of antidiagonals of the Sierpinski carpet A153490.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 6, 6, 4, 8, 10, 8, 13, 14, 10, 14, 13, 8, 14, 16, 12, 18, 18, 12, 16, 14, 8, 16, 20, 16, 26, 28, 20, 28, 26, 16, 29, 34, 26, 40, 41, 28, 38, 34, 20, 34, 38, 28, 41, 40, 26, 34, 29, 16, 30, 36, 28, 44, 46, 32, 44, 40, 24, 42, 48, 36, 54, 54, 36, 48, 42, 24

Offset: 1

M. F. Hasler, Oct 24 2017

Also, sums of digits of terms of A292688, or Hamming weights of terms of A292689. See there or A153490 for definition / construction of the Sierpiski carpet.

Rémy Sigrist, Table of n, a(n) for n = 1..19683

Cf. A153490, A292688, A292689, A000120, A007953, A000217.

A293974[i_]:=With[{a=Nest[ArrayFlatten[{{#,#,#},{#,0,#},{#,#,#}}]&,{{1}},i]},Array[Total[Diagonal[a,#]]&,3^i,1-3^i]];A293974[5] (* Generates 3^5 terms *) (* Paolo Xausa, May 14 2023 *)

PARI
```
A293974(n,A=Mat(1))={while(#A
				
```

a(n) = A007953(A292688(n)) = A000120(A292689(n)) = sum(k=1..n, A153490(n,k)), considering A153490 as triangle; could also be indexed as matrix (m,n = 1,...,oo) or "flattened" (linearized) using A000217.

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A153490 Sierpinski carpet, read by antidiagonals.

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A292688 Antidiagonals of the Sierpinski carpet (as binary numbers).

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A293974 Row sums of antidiagonals of the Sierpinski carpet A153490.

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