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Search: id:A068490
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| A068490 |
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In the Ana sequence, use (the ungrammatical) "a n" instead of "an n" in describing n's. That is, begin with the letter "a". Generate the next term by using the indefinite article as appropriate, but using "a n" instead of "an n". E.g. "an a", then "an a, a n, an a" etc. Assign a=1, n=2. |
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+0
1
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| 1, 121, 12112121, 121121211211212112121, 1211212112112121121211211212112112121121211211212112121, 12112121121121211212112112121121121211212112112121121211211212112112121121211211\ 2121121121211212112112121121211211212112112121121211211212112121
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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For proofs of the following assertions, see the link to the paper "Ana's Golden Fractal". Let A(n), N(n) denote the number of 1's and the number of 2's in a(n). Then for n > 1, A(n), N(n) are consecutive Fibonacci numbers: A(n) = F(2n-1), N(n) = F(2n-2), where F(k) denotes the k-th Fibonacci number. Hence lim_{n} A(n)/N(n) = phi, the golden ratio.
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 using the golden Ana sequence has fractal dimension = 1.
A fixed point of the morphism 1 -> 121, 2 -> 12, starting from a(1) = 1. See A003842.
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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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Pe, J., Ana's Golden Fractal
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
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MATHEMATICA
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f[n_] := FromDigits[ Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {1, 2}}] &, {1}, n]]; Table[ f[n], {n, 0, 5}] (from Robert G. Wilson v Mar 05 2005)
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CROSSREFS
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Cf. A060032.
Sequence in context: A123179 A136094 A053885 this_sequence A077735 A068121 A013859
Adjacent sequences: A068487 A068488 A068489 this_sequence A068491 A068492 A068493
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KEYWORD
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nonn
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AUTHOR
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Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Mar 11 2002
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