cp's OEIS Frontend

A156253 Least k such that A054353(k) >= n.

%I A156253 #41 Jul 07 2023 06:14:00
%S A156253 1,2,2,3,3,4,5,6,6,7,8,8,9,9,10,11,11,12,12,13,14,15,15,16,17,18,18,
%T A156253 19,19,20,21,21,22,23,24,24,25,26,26,27,27,28,29,30,30,31,32,33,33,34,
%U A156253 35,35,36,36,37,38,38,39,39,40,41,42,42,43,44,44,45,45,46,47,47,48,49,50
%N A156253 Least k such that A054353(k) >= n.
%C A156253 a(n)=1 plus the number of symbol changes in the first n terms of A000002. - Jean-Marc Fedou and _Gabriele Fici_, Mar 18 2010
%C A156253 From _N. J. A. Sloane_, Nov 12 2018: (Start)
%C A156253 This seems to be A001462 rewritten so the run lengths are given by A000002. The companion sequence, A000002 rewritten so the run lengths are given by A001462, is A321020.
%C A156253 Note that Kolakoski's sequence A000002 and Golomb's sequence A001462 have very similar definitions, although the asymptotic behavior of A001462 is well-understood, while that of A000002 is a mystery. The asymptotic behavior of the two hybrids A156253 and A321020 might be worth investigating. (End)
%C A156253 To expand upon N. J. A. Sloane's comments, it's worth noting that Golomb's sequence has a formula from Colin Mallows: g(n) = g(n-g(g(n-1))) + 1, which closely resembles a(n) = a(n-gcd(a(a(n-1)),2)) + 1. - _Jon Maiga_, May 16 2023
%H A156253 Jon Maiga, <a href="/A156253/b156253.txt">Table of n, a(n) for n = 1..10000</a>
%H A156253 J. M. Fedou and G. Fici, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL13/Fici/fici.html">Some remarks on differentiable sequences and recursivity</a>, Journal of Integer Sequences 13(3): Article 10.3.2 (2010).
%H A156253 Jon Maiga, <a href="/A156253/a156253.pdf">A Recurrence Related to the Kolakoski Sequence</a>
%H A156253 N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides</a> (Mentions this sequence)
%F A156253 Conjecture: a(n) should be asymptotic to 2n/3.
%F A156253 Length of n-th run of the sequence = A000002(n). - _Benoit Cloitre_, Feb 19 2009
%F A156253 Conjecture: a(n) = (a(a(n-1)) mod 2) + a(n-2) + 1. - _Jon Maiga_, Dec 09 2021
%F A156253 a(n) = a(n-gcd(a(a(n-1)), 2)) + 1. - _Jon Maiga_, May 16 2023
%t A156253 a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 80}, {i, 1, a2[[n]]}]; a3 = Accumulate[a2]; a[1] = 1; a[n_] := a[n] = For[k = a[n - 1], True, k++, If[a3[[k]] >= n, Return[k]]]; Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Jun 18 2013 *)
%t A156253 a[1] = 1;
%t A156253 a[n_]:=a[n]=a[n-GCD[a[a[n - 1]], 2]]+1
%t A156253 Array[a, 100] (* _Jon Maiga_, May 16 2023 *)
%Y A156253 Cf. A000002, A001462, A054353, A156351, A321020.
%K A156253 nonn
%O A156253 1,2
%A A156253 _Benoit Cloitre_, Feb 07 2009