Search: id:A068490
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%I A068490
%S A068490 1,121,12112121,121121211211212112121,
%T A068490 1211212112112121121211211212112112121121211211212112121,
%U A068490 1211212112112121121211211212112112121121211211212112121121121211211212112121121121211211212112121121121211212\
               11211212112112121121211211212112121
%N A068490 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.
%C A068490 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.
%C A068490 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.
%C A068490 A fixed point of the morphism 1 -> 121, 2 -> 12, starting from a(1) = 
               1. See A003842.
%D A068490 C. Pickover, Wonders of Numbers, Chap. 69 "An A?", Oxford University 
               Press, NY, 2001, p. 167-171.
%H A068490 Pe, J., <a href="http://www.geocities.com/windmill96/anagoldenfractal/
               anagoldenfractal.html">Ana's Golden Fractal</a>
%H A068490 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind 
               and Meaning," <a href="http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an\
               =0983.00008&format=complete">Zentralblatt review</a>
%t A068490 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)
%Y A068490 Cf. A060032.
%Y A068490 Sequence in context: A123179 A136094 A053885 this_sequence A077735 A068121 
               A013859
%Y A068490 Adjacent sequences: A068487 A068488 A068489 this_sequence A068491 A068492 
               A068493
%K A068490 nonn
%O A068490 1,2
%A A068490 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Mar 11 2002

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