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A333363 Horizontal visibility sequence at the onset of chaos in the 3-period cascade.

%I A333363 #46 Mar 08 2024 09:44:48
%S A333363 3,2,5,3,2,7,3,2,5,3,2,9,3,2,5,3,2,7,3,2,5,3,2,11,3,2,5,3,2,7,3,2,5,3,
%T A333363 2,9,3,2,5,3,2,7,3,2,5,3,2,13,3,2,5,3,2,7,3,2,5,3,2,9,3,2,5,3,2,7,3,2,
%U A333363 5,3,2,11,3,2,5,3,2,7,3,2,5,3,2,9,3,2,5,3,2,7,3,2,5,3,2,15
%N A333363 Horizontal visibility sequence at the onset of chaos in the 3-period cascade.
%C A333363 This sequence represents the horizontal visibility of the points of the chaotic time series at the onset of chaos in the 3-period cascade of the logistic (unimodal) map.
%C A333363 Observation: if the sequence is written as a table array with six columns read by rows we have that, at least for the first 16 rows, the n-th row is "3, 2, 5, 3, 2" together with (6 + A037227(n)), see the example. - _Omar E. Pol_, Mar 16 2020
%H A333363 Michael De Vlieger, <a href="/A333363/b333363.txt">Table of n, a(n) for n = 1..10000</a>
%H A333363 Juan C. Nuño and Francisco J. Muñoz, <a href="https://arxiv.org/abs/2002.10423">Universal visibility patterns of unimodal maps</a>, arXiv:2002.10423 [nlin.CD], 2020.
%H A333363 Juan C. Nuño and Francisco J. Muñoz, <a href="https://doi.org/10.1063/5.0006652">Universal visibility patterns of unimodal maps</a>, Chaos, 30, 063105 (2020).
%H A333363 Juan Carlos Nuño and Francisco J. Muñoz, <a href="https://arxiv.org/abs/2009.14629">On the ubiquity of the ruler sequence</a>, arXiv:2009.14629 [math.HO], 2020.
%H A333363 Juan Carlos Nuño and Francisco J. Muñoz, <a href="https://doi.org/10.3390/math12050742">The Ruler Sequence Revisited: A Dynamic Perspective</a>, Mathematics (2024) Vol. 12, No. 5, 742.
%F A333363 Conjectured: a(n) = 2*A007814(n/3) + 5 if 3|n and a(n) = 4 - (n mod 3) otherwise. - _Giovanni Resta_, Mar 16 2020
%e A333363 From _Omar E. Pol_, Mar 16 2020: (Start)
%e A333363 Written as a table with six columns read by rows:
%e A333363   3, 2, 5, 3, 2,  7;
%e A333363   3, 2, 5, 3, 2,  9;
%e A333363   3, 2, 5, 3, 2,  7;
%e A333363   3, 2, 5, 3, 2, 11;
%e A333363   3, 2, 5, 3, 2,  7;
%e A333363   3, 2, 5, 3, 2,  9;
%e A333363   3, 2, 5, 3, 2,  7;
%e A333363   3, 2, 5, 3, 2, 13;
%e A333363   3, 2, 5, 3, 2,  7;
%e A333363   3, 2, 5, 3, 2,  9;
%e A333363   3, 2, 5, 3, 2,  7;
%e A333363   3, 2, 5, 3, 2, 11;
%e A333363   3, 2, 5, 3, 2,  7;
%e A333363   3, 2, 5, 3, 2,  9;
%e A333363   3, 2, 5, 3, 2,  7;
%e A333363   3, 2, 5, 3, 2, 15;
%e A333363 (End)
%t A333363 L[n_] := L[n] = Block[{s = {3, 2, 2*n+3}}, Do[s = Join[L[i], s], {i, n-1}]; s]; L[6] (* _Giovanni Resta_, Mar 16 2020 *)
%o A333363 (R)
%o A333363 visibsuc3 <- function(n){
%o A333363     suc <- c(3,2, 2*(n+1)+1)
%o A333363     if(n>1){
%o A333363     for(i in 1:(n-1)){
%o A333363     suc <- c(visibsuc3(i), suc)
%o A333363     }
%o A333363    }
%o A333363    return(suc)
%o A333363   }
%Y A333363 Cf. A007814, A037227.
%K A333363 nonn,tabf
%O A333363 1,1
%A A333363 _Francisco J. Muñoz_ and _Juan Carlos Nuño_, Mar 16 2020