A068490 Fixed point of the morphism 1 -> 121, 2 -> 12, starting from 1.

1, 121, 12112121, 121121211211212112121, 1211212112112121121211211212112112121121211211212112121, 121121211211212112121121121211211212112121121121211212112112121121121211212112112121121121211212112112121121211211212112112121121211211212112121

Offset: 1

Author

Joseph L. Pe, Mar 11 2002

Keywords

• nonn

Comments

Previous name was: 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.

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.

Fixed point of the morphism 1 -> 121, 2 -> 12, starting from a(1) = 1. See A003842.

References

• C. Pickover, Wonders of Numbers, Chap. 69 "An A?", Oxford University Press, NY, 2001, pp. 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. A060032.

Programs

Mathematica

f[n_] := FromDigits[ Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {1, 2}}] &, {1}, n]]; Table[ f[n], {n, 0, 5}] (* Robert G. Wilson v, Mar 05 2005 *)

Extensions

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