

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
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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 nth iteration, the upperleft 3^n X 3^n squares will always remain the same. Therefore this sequence, which reads these by antidiagonals, is welldefined.
The nth 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



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, ...


MATHEMATICA

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


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!=2j%3!=2))); sum(k=0, n1, if(A[k+1, nk], 10^k))}


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



