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A333363
Horizontal visibility sequence at the onset of chaos in the 3-period cascade.
1
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
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
Sequence in context: A361317 A361783 A194736 * A130299 A143309 A206562
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved