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A292688 Antidiagonals of the Sierpinski carpet (as binary numbers). 5
1, 11, 101, 1111, 11111, 101101, 1110111, 11100111, 101000101, 1111001111, 11111011111, 101101101101, 1111111111111, 11111111111111, 101101101101101, 1110111111110111, 11100111111100111, 101000101101000101, 1111001110111001111, 11111011100111011111, 101101101000101101101 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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,...)

LINKS

Table of n, a(n) for n=1..21.

Eric Weisstein's World of Mathematics, Sierpinski Carpet.

Wikipedia, Sierpinski carpet.

EXAMPLE

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, ...

PROG

(PARI) A292688(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))); sum(k=0, n-1, if(A[k+1, n-k], 10^k))}

CROSSREFS

Cf. A153490, A292689.

Sequence in context: A265427 A284480 A290660 * A286519 A088292 A135563

Adjacent sequences:  A292685 A292686 A292687 * A292689 A292690 A292691

KEYWORD

nonn

AUTHOR

M. F. Hasler, Oct 23 2017

STATUS

approved

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Last modified April 25 16:59 EDT 2019. Contains 322461 sequences. (Running on oeis4.)