The Ruler Sequence Revisited: A Dynamic Perspective

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors present a nice overview of the ruler sequence. It seems that this combinatorial construction appears frequently in several apparently disconnected areas of mathematics and quantitative sciences. The authors have made an effort to give a recollection of these areas, and attempted to reconcile these apparent coincidences in terms of the sequence recursiveness and self-containing. I think this is a nice contribution that could help to bridge different topics between dynamical systems and combinatorics, and I therefore recommend publication.

Below I provide a few comments that might help to slightly improve the presentation.

-- The relation between horizontal visibility graphs throughout the Feigenbaum scenario (called Feigenbaum graphs) and polygon constructions was made explicit in [Flanagan et al, On the spectral of Feigenbaum graphs, J. Phys. A 53 (2019)] (see Fig.5 in that paper). The authors could perhaps discuss this.

-- Furthermore, the relation between these Feigenbaum graphs and the so-called Farey graphs [Zhang et al, Counting spanning trees in a small-world Farey graph, Physica A 391 (2012)], which are intimately related to Stern-Brocot trees, was made explicit in a footnote in [Calero-Sanz et al, Haros graphs: an exotic representation of real numbers. Journal of Complex Networks 10, 5 (2022)].

 

 

Author Response

\textbf{Reviewer 1.}

 

R11. The relation between horizontal visibility graphs throughout the Feigenbaum scenario (called Feigenbaum graphs) and polygon constructions was made explicit in [Flanagan et al, On the spectral of Feigenbaum graphs, J. Phys. A 53 (2019)] (see Fig.5 in that paper). The authors could perhaps discuss this.

 

As suggested by this reviewer, we have included a new paragraph in the last section to highlight the meaning of the ruler sequence in the $n$-gons generated by side bisection, pointing out that this process was previously suggested in Flanagan et al., On the spectral of Feigenbaum graphs, J. Phys. A 53 (2019). Explicitly, this is the paragraph added in the new version of the paper:

 

\textit{“This generative process that underlies the ruler sequence is also evident in the sequence of vertex indices of the $n$-gons generated by size bisection. For each $2^n$-gon generated by bisecting sides from the $2^{n-1}$-gon, where $n=1,2,\ldots$, a sequence of length $2^n-1$ is associated. This sequence contains the number of previous polygons to which each vertex belongs, ordered from the left of the southernmost vertex and proceeding clockwise (see Fig.5). It should be noted that a similar duplication process is suggested in [Flanagan et al. 2019], although with a different motivation. Remarkably, what we have shown is that, in the limit of $n$ tending to infinity when the $n$-gons tend to the circumference, the ruler sequence describes the indices of the infinite numbers that form the circumference.”}

 

A reference to Feigenbaum graphs has also been made explicit in the introduction:

 

\textit{“The bifurcation diagram has been reinterpreted by applying the horizontal visibility algorithm [Luque et al., 2011], obtaining a special type of graphs denominated {\it Feigenbaum graphs} [Flanagan et al., 2019].”}

 

R12. Furthermore, the relation between these Feigenbaum graphs and the so-called Farey graphs [Zhang et al, Counting spanning trees in a small-world Farey graph, Physica A 391 (2012)], which are intimately related to Stern-Brocot trees, was made explicit in a footnote in [Calero-Sanz et al, Haros graphs: an exotic representation of real numbers. Journal of Complex Networks 10, 5 (2022)].

 

As suggested by the reviewer, we have introduced a couple of sentences highlighting the relationship between the ruler sequence and other sequences, in particular with Farey fractions:

 

\textit{“Notably, the ruler sequence is related to other well-known integer sequences [Kimberling, 2007]. Especially significant in this context is its relationship with Farey fractions through concepts like continued fractions and structures like the Stern-Brocot tree [Calero-Sanz et al., 2022].”}

 

Where a new reference to the paper by Calero-Sanz et al., 2022, is also included.

 

Reviewer 2 Report

Comments and Suggestions for Authors

This work is devoted to the study of some interesting recurrence sequences. It contains no serious mathematical result and no theorems at all, but is well and lively written and contains some nice pictures, diagrams, etc. It can be viewed as a popularization of the subject which is supported by many references. I like the paper: I believe it deserves to be published. 

Author Response

\textbf{Reviewer 2.}

 

No further requirements!

 

Reviewer 3 Report

Comments and Suggestions for Authors

This paper provides several mathematical observations on the ruler function encoded in the OEIS as A001511.

The authors describe the presence of such a function in the demographic discrete dynamical automaton, the middle interval cantor set, and the sequence at the accumulation point of the Feigenbaum cascade.

The paper can be considered a survey paper dealing with the integer sequence. However, more information than provided is needed to consider it a document giving complete details on the sequence.

The authors could provide more details on some of their statements, e.g., "It is established that the total lengths (Cantor intervals) tend to 1. " Furthermore, they could prove that the sequence associated with polygons is precisely the integer sequence A001511. The same is true for the sequence related to the accumulation point of the Feigenbaum cascade.

 

Author Response

\textbf{Reviewer 3.}

 

R31. The paper can be considered a survey paper dealing with the integer sequence. However, more information than provided is needed to consider it a document giving complete details on the sequence.

 

We would like to emphasize that our focus is not on producing a survey paper about the ruler sequence. Instead, our aim is to highlight the dynamic aspects underlying its generation. However, we have included a short paragraph in the last section that relates this sequence to other classical sequences.

 

\textit{“Notably, the ruler sequence is related to other well-known integer sequences [Kimberling, 2007]. Especially significant in this context is its relationship with Farey fractions through concepts like continued fractions and structures like the Stern-Brocot tree [Calero-Sanz et al., 2022].”}

 

R32. The authors could provide more details on some of their statements, e.g., "It is established that the total lengths (Cantor intervals) tend to 1". Furthermore, they could prove that the sequence associated with polygons is precisely the integer sequence A001511. The same is true for the sequence related to the accumulation point of the Feigenbaum cascade.

 

As suggested by the reviewer, we have included a proof that the total length of the middle intervals (the complementary of the Cantor set) tends to 1 as $n$ tends to infinity.

 

\textit{“In general, it can be proven that:}

 

\begin{equation}

L_n = \sum_{k=1}^n \, 2^{k-1} \, \frac{1}{3^k} = \frac{1}{2} \, \sum_{k=1}^n \, \left(\frac{2}{3}\right)^k = \frac{1}{2} \, \left( 3 \, \left(1 - \left(\frac{2}{3} \right)^n\right) - 1 \right)

\end{equation}

 

\textit{This sequence of total lenghts $\{L_n\}$ tends to 1 as $n$ tends to infinity. Note that this limit coincides with the total length of the complementary of the Cantor set [Vallin, 2013].”}

 

To demonstrate the equivalence with the ruler sequence, we have included a function written in R, referred to in the text as VisPattern. This function generates the forward visibility patterns for the Feigenbaum cascade, specifically as period doubling occurs. Remarkably, the sequence generated by VisPattern is identical to the algorithm written by Caroli, which generates the Ruler sequence, starting from the largest value for each $n$.

 

\textit{“This construction can be algorithmically performed by the following recursive function: }

 

\begin{verbatim}

VisPattern <- function(n){

if(n==0){

return(1)

}else{

s <- c()

for(i in 1:n){

s <- c(s, VisPattern(i-1))

}

return(c(n+1, s))

}

}

\end{verbatim}

 

Note that this function yields the same sequence as the iterative script proposed by Caroli to generate the ruler sequence, as explicitly depicted in section 2. However, it's worth noting that VisPattern starts from the middle largest value for each $n$.”

 

This would prove that both sequences are the same, except for the starting number in VisPattern, which is the largest one for each $n$.

 

The similar constructive process by side bisection yields the sequence of vertex indices in the $2^n$-gons then, coinciding with the ruler sequence.

 

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

Dear authors, the manuscript has been improved. In my opinion, it deserves to be published in MATHEMATICS