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

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

Offset: 1

Author

M. F. Hasler, Oct 23 2017

Keywords

• nonn

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,...)

Examples

			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, ...
		

Links

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

Crossrefs

Cf. A153490, A292689.

Programs

Mathematica

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