A060032 Fixed point of the morphism 1 -> 121, 2 -> 122, starting from 1.

1, 121, 121122121, 121122121121122122121122121, 121122121121122122121122121121122121121122122121122122121122121121122122121122121

Offset: 0

Author

Jason Earls, Mar 17 2001

Keywords

• nonn

Comments

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.

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

Examples

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

References

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

Links

• Joseph L. Pe, Ana's Golden Fractal, Fractals, Vol. 11, No. 4 (2003) 309-313.
• C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Crossrefs

Cf. A060236.

Programs

Mathematica

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 *)

Formula

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.

Extensions

Additional comments from Joseph L. Pe, Mar 11 2002

More descriptive name from comment, Joerg Arndt, Jan 23 2024