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A153490
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Sierpinski carpet, read by antidiagonals.
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9
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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
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OFFSET
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1,1
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COMMENTS
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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, ...}.
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LINKS
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EXAMPLE
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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},
...
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MATHEMATICA
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<< MathWorld`Fractal`; fractal = SierpinskiCarpet;
a = fractal[4]; Table[Table[a[[m]][[n - m + 1]], {m, 1, n}], {n, 1, 12}];
Flatten[%]
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PROG
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(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
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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