|
%I #15 May 19 2023 04:54:40
%S 1,11,101,1111,11111,101101,1110111,11100111,101000101,1111001111,
%T 11111011111,101101101101,1111111111111,11111111111111,
%U 101101101101101,1110111111110111,11100111111100111,101000101101000101,1111001110111001111,11111011100111011111,101101101000101101101
%N Antidiagonals of the Sierpinski carpet (as binary numbers).
%C Concatenation of the terms in the rows of A153490.
%C 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 The n-th term a(n) has n digits. See A292689 for the decimal value of a(n) considered as binary number.
%C 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 Paolo Xausa, <a href="/A292688/b292688.txt">Table of n, a(n) for n = 1..729</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiCarpet.html">Sierpinski Carpet</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sierpinski_carpet">Sierpinski carpet</a>.
%e The Sierpinski carpet matrix A153490 reads
%e 1 1 1 1 1 1 1 1 1...
%e 1 0 1 1 0 1 1 0 1...
%e 1 1 1 1 1 1 1 1 1...
%e 1 1 1 0 0 0 1 1 1...
%e 1 0 1 0 0 0 1 0 1...
%e 1 1 1 0 0 0 1 1 1...
%e 1 1 1 1 1 1 1 1 1...
%e 1 0 1 1 0 1 1 0 1...
%e 1 1 1 1 1 1 1 1 1...
%e (...)
%e 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[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 (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 Cf. A153490, A292689.
%K nonn
%O 1,2
%A _M. F. Hasler_, Oct 23 2017
|
