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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 Paolo Xausa, Table of n, a(n) for n = 1..729 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

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