A078880 The sequence starting with 2 that equals its own run length sequence.

2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1

Offset: 1

Author

Clark Kimberling, Dec 11 2002

Keywords

• nonn

Comments

It is an unsolved problem to show that the density of 1's is equal to 1/2.

The sequence can be generated by starting with 22 and applying the block-substitution rules 22 -> 2211, 21 -> 221, 12 -> 211, 11 -> 21. (Lagarias)

Examples

			Start with 2, which generates 22 (so that the first run length is 2); then 22 generates 2211 (so that the first two run lengths are 2 and 2); then 2211 generates 221121 and so on.
		

References

• M. S. Keane, Ergodic theory and subshifts of finite type, Chap. 2 of T. Bedford et al., eds., Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford, 1991, esp. p. 50.

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• J.-M. Fedou, G. Fici, Some remarks on differentiable sequences and recursivity, JIS 13 (2010) # 10.3.2.

Crossrefs

See A000002, this sequence prepended with 1, for properties, formulas, references, links, programs, etc.

Programs

Mathematica

seed = {2, 1}; w = {}; i = 1; Do[w = Join[w, Array[seed[[Mod[i - 1, Length[seed]] + 1]] &, If[i > Length[w], seed, w][[i]]]]; i++, {n, 70}]; w (* Ivan Neretin, Apr 02 2015 *)

Formula

a(n) = k(n+1), where k=A000002, the Kolakoski sequence.