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

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, 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, 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

Offset

1,1

Comments

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.

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Juan C. Nuño and Francisco J. Muñoz, Universal visibility patterns of unimodal maps, arXiv:2002.10423 [nlin.CD], 2020.

Juan C. Nuño and Francisco J. Muñoz, Universal visibility patterns of unimodal maps, Chaos, 30, 063105 (2020).

Juan Carlos Nuño and Francisco J. Muñoz, On the ubiquity of the ruler sequence, arXiv:2009.14629 [math.HO], 2020.

Juan Carlos Nuño and Francisco J. Muñoz, The Ruler Sequence Revisited: A Dynamic Perspective, Mathematics (2024) Vol. 12, No. 5, 742.

Formula

Conjectured: a(n) = 2*A007814(n/3) + 5 if 3|n and a(n) = 4 - (n mod 3) otherwise. - Giovanni Resta, Mar 16 2020

Example

From Omar E. Pol, Mar 16 2020: (Start)
Written as a table with six columns read by rows:
  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, 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, 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;
(End)

Mathematica

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 *)

Prog

(R)
visibsuc3 <- function(n){
    suc <- c(3, 2, 2*(n+1)+1)
    if(n>1){
    for(i in 1:(n-1)){
    suc <- c(visibsuc3(i), suc)
    }
   }
   return(suc)
  }

Crossrefs

Cf. A007814, A037227.

Sequence in context: A361317 A361783 A194736 * A130299 A143309 A206562

Adjacent sequences: A333360 A333361 A333362 * A333364 A333365 A333366

Keyword

nonn,tabf

Author

Francisco J. Muñoz and Juan Carlos Nuño, Mar 16 2020

Status

approved