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A078880 The sequence starting with 2 that equals its own run length sequence. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A278387 A074293 A013949 * A000002 A074295 A236479

Adjacent sequences:  A078877 A078878 A078879 * A078881 A078882 A078883

KEYWORD

nonn

AUTHOR

Clark Kimberling, Dec 11 2002

STATUS

approved

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Last modified August 19 08:57 EDT 2017. Contains 290794 sequences.