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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
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, ...
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 *)
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
Sequence in context: A265427 A284480 A290660 * A286519 A088292 A135563
KEYWORD
nonn
AUTHOR
M. F. Hasler, Oct 23 2017
STATUS
approved

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Last modified May 21 09:41 EDT 2024. Contains 372733 sequences. (Running on oeis4.)