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

%I A292688 #16 Feb 16 2025 08:33:51
%S A292688 1,11,101,1111,11111,101101,1110111,11100111,101000101,1111001111,
%T A292688 11111011111,101101101101,1111111111111,11111111111111,
%U A292688 101101101101101,1110111111110111,11100111111100111,101000101101000101,1111001110111001111,11111011100111011111,101101101000101101101
%N A292688 Antidiagonals of the Sierpinski carpet (as binary numbers).
%C A292688 Concatenation of the terms in the rows of A153490.
%C A292688 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.
%C A292688 The n-th term a(n) has n digits. See A292689 for the decimal value of a(n) considered as binary number.
%C A292688 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,...)
%H A292688 Paolo Xausa, <a href="/A292688/b292688.txt">Table of n, a(n) for n = 1..729</a>
%H A292688 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiCarpet.html">Sierpinski Carpet</a>.
%H A292688 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sierpinski_carpet">Sierpinski carpet</a>.
%e A292688 The Sierpinski carpet matrix A153490 reads
%e A292688    1 1 1 1 1 1 1 1 1...
%e A292688    1 0 1 1 0 1 1 0 1...
%e A292688    1 1 1 1 1 1 1 1 1...
%e A292688    1 1 1 0 0 0 1 1 1...
%e A292688    1 0 1 0 0 0 1 0 1...
%e A292688    1 1 1 0 0 0 1 1 1...
%e A292688    1 1 1 1 1 1 1 1 1...
%e A292688    1 0 1 1 0 1 1 0 1...
%e A292688    1 1 1 1 1 1 1 1 1...
%e A292688    (...)
%e A292688 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, ...
%t A292688 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 *)
%o A292688 (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))}
%Y A292688 Cf. A153490, A292689.
%K A292688 nonn
%O A292688 1,2
%A A292688 _M. F. Hasler_, Oct 23 2017