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

%I A001083 #43 Feb 16 2025 08:32:22
%S A001083 1,2,2,3,5,7,10,15,23,34,50,75,113,170,255,382,574,863,1293,1937,2903,
%T A001083 4353,6526,9789,14688,22029,33051,49577,74379,111580,167388,251090,
%U A001083 376631,564932,847376,1271059,1906628,2859984
%N A001083 Length of one version of Kolakoski sequence {A000002(i)} at n-th growth stage.
%H A001083 Konstantinos Lambropoulos and Constantinos Simserides, <a href="https://arxiv.org/abs/1808.04764">Spectral, localization and charge transport properties of periodic, aperiodic and random binary sequences</a>, arXiv:1808.04764 [cond-mat.soft], 2018.
%H A001083 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KolakoskiSequence.html">Kolakoski Sequence</a>
%F A001083 Conjecture: a(n) is asymptotic to c*(3/2)^n where c=0.5819.... - _Benoit Cloitre_, Jun 01 2004
%F A001083 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]
%F A001083 Equivalently, a(n) = A054353(a(n-1)) for n>3. - _Jon Maiga_, Jul 10 2022
%e A001083 /* generate sequence of sequences by recursion using next1() ( origin 1 ) */
%e A001083 v=[2]; for(n=1,8,p1(v); print1(" -> "); v=next1(v))
%e A001083 2 -> 11 -> 12 -> 122 -> 12211 -> 1221121 -> 1221121221 -> 122112122122112 ->
%e A001083 v=[2]; for(n=1,8,print1(length(v)); print1(","); v=next1(v)) gives: 1,2,2,3,5,7,10,15,
%o A001083 (PARI) /* generate sequence starting at 1 given run length sequence */
%o A001083 next1(v)=local(w); w=[]; for(n=1,length(v), for(i=1,v[n],w=concat(w,2-n%2))); w
%o A001083 /* print a number or sequence recursively with no commas */
%o A001083 p1(v)=if(type(v)!="t_VEC",print1(v), for(n=1,length(v),p1(v[n])))
%Y A001083 Cf. A000002, A042942, A054353.
%K A001083 nonn
%O A001083 1,2
%A A001083 _N. J. A. Sloane_, _Simon Plouffe_
%E A001083 Corrected by and better description from _Michael Somos_, May 05 2000