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A060032
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Fixed point of the morphism 1 -> 121, 2 -> 122, starting from 1.
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4
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OFFSET
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0,2
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COMMENTS
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Previous name was: Ana sequence.
Let A(n), N(n) denote the number of 1's and the number of 2's in a(n). Then A(n) = (3^(k-1) + 1)/2, N(n) = (3^(k-1) - 1)/2. Hence lim_{n} A(n)/N(n) = 1.
In "Wonders of Numbers", Pickover considers a "fractal bar code" constructed from the Ana sequence. Start with a segment I of fixed length; at stage n, evenly subdivide I into as many non-overlapping closed intervals as there are letters in the n-th term of the Ana sequence; then shade the intervals corresponding to a's. It can be shown that a fractal set defined from this construction has fractal dimension = 1.
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REFERENCES
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C. Pickover, Wonders of Numbers, Chap. 69 "An A?", Oxford University Press, NY, 2001, p. 167-171.
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LINKS
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C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
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FORMULA
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Begin with the letter "a". Generate next term by using the indefinite article as appropriate, e.g., "an a", then "an a, an n, an a" etc. Assign a=1, n=2.
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EXAMPLE
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a(2) = ana = 121, a(3) = ana ann ana = 121122121.
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MATHEMATICA
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f[n_] := FromDigits[ Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {1, 2, 2}}] &, {1}, n]]; Table[ f[n], {n, 0, 4}] (* Robert G. Wilson v, Mar 05 2005 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More descriptive name from comment, Joerg Arndt, Jan 23 2024
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STATUS
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approved
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