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

1,1

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)

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.

Formula

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A074293 A013949 A331349 * A000002 A074295 A331348

Adjacent sequences: A078877 A078878 A078879 * A078881 A078882 A078883

Keyword

nonn

Author

Clark Kimberling, Dec 11 2002

Status

approved