0,2

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.

A fixed point of the morphism 1 -> 121, 2 -> 122, starting from a(1) = 1. See A060236. - from Robert G. Wilson v Mar 05 2005)

C. Pickover, Wonders of Numbers, Chap. 69 "An A?", Oxford University Press, NY, 2001, p. 167-171.

Pe, J., Ana's Golden Fractal

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

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.

a(2) = ana = 121, a(3) = ana ann ana = 121122121.

f[n_] := FromDigits[ Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {1, 2, 2}}] &, {1}, n]]; Table[ f[n], {n, 0, 4}] (from Robert G. Wilson v Mar 05 2005)

Sequence in context: A077735 A068121 A013859 this_sequence A109643 A080826 A043646

Adjacent sequences: A060029 A060030 A060031 this_sequence A060033 A060034 A060035

nonn

Jason Earls (zevi_35711(AT)yahoo.com), Mar 17 2001

Additional comments from Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Mar 11 2002