A001083 Length of one version of Kolakoski sequence {A000002(i)} at n-th growth stage.

1, 2, 2, 3, 5, 7, 10, 15, 23, 34, 50, 75, 113, 170, 255, 382, 574, 863, 1293, 1937, 2903, 4353, 6526, 9789, 14688, 22029, 33051, 49577, 74379, 111580, 167388, 251090, 376631, 564932, 847376, 1271059, 1906628, 2859984

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Examples

			/* generate sequence of sequences by recursion using next1() ( origin 1 ) */
v=[2]; for(n=1,8,p1(v); print1(" -> "); v=next1(v))
2 -> 11 -> 12 -> 122 -> 12211 -> 1221121 -> 1221121221 -> 122112122122112 ->
v=[2]; for(n=1,8,print1(length(v)); print1(","); v=next1(v)) gives: 1,2,2,3,5,7,10,15,
		

Links

• Konstantinos Lambropoulos and Constantinos Simserides, Spectral, localization and charge transport properties of periodic, aperiodic and random binary sequences, arXiv:1808.04764 [cond-mat.soft], 2018.
• Eric Weisstein's World of Mathematics, Kolakoski Sequence

Crossrefs

Cf. A000002, A042942, A054353.

Programs

PARI

/* generate sequence starting at 1 given run length sequence */
next1(v)=local(w); w=[]; for(n=1,length(v), for(i=1,v[n],w=concat(w,2-n%2))); w
/* print a number or sequence recursively with no commas */
p1(v)=if(type(v)!="t_VEC",print1(v), for(n=1,length(v),p1(v[n])))

Formula

Conjecture: a(n) is asymptotic to c*(3/2)^n where c=0.5819.... - Benoit Cloitre, Jun 01 2004

For n >= 1, a(n+3) = S^n(2) where S(n) = A054353(n) and S^k(2) = S(S^(k-1)(2)). - Benoit Cloitre, Feb 24 2009 [adjusted to match sequence offset by Jon Maiga, Jul 27 2022]

Equivalently, a(n) = A054353(a(n-1)) for n>3. - Jon Maiga, Jul 10 2022

Extensions

Corrected by and better description from Michael Somos, May 05 2000