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

1,2

Links

Table of n, a(n) for n=1..38.

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

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

Example

/* 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,

Prog

(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])))

Crossrefs

Cf. A000002, A042942, A054353.

Sequence in context: A173693 A058278 A097333 * A173696 A120412 A022864

Adjacent sequences: A001080 A001081 A001082 * A001084 A001085 A001086

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

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

Status

approved