The Perrin Sequence

Many years ago, in the bookstore at the bottom of SOGO in Pondok Indah Mall (Jakarta), I acquired a book by David Wells titled Prime Numbers: The Most Mysterious Figures in Math. I still have the physical copy but I also now have an electronic copy. On page 103, there is the information shown in Figure 1:

Figure 1

It so happens that the number representing my diurnal age today, 26149, is a member of this "very generalized Fibonacci sequence" and is in fact OEIS A050443:

A050443

a(0)=4, a(1)=0, a(2)=0, a(3)=3; thereafter a(n) = a(n-3) + a(n-4).

4, 0, 0, 3, 4, 0, 3, 7, 4, 3, 10, 11, 7, 13, 21, 18, 20, 34, 39, 38, 54, 73, 77, 92, 127, 150, 169, 219, 277, 319, 388, 496, 596, 707, 884, 1092, 1303, 1591, 1976, 2395, 2894, 3567, 4371, 5289, 6461, 7938, 9660, 11750, 14399, 17598, 21410, 26149

I was alerted to this connection to the book thanks to an OEIS reference that also remarks that the sequence is related to the Perrin sequence that also has the property that a(\(p\)) is divisible by \(p\) for primes \(p\). So what is the Perrin sequence? WolframMathWorld defines it as:$$P(n)=P(n-2)+P(n-3) \text{ where }P(0)=3,P(1)=0,P(2)=2$$Unusually, the Wolfram article also displays a cartoon (see Figure 2):

Figure 2: The above cartoon (Amend 2005) shows an unconventional sports
application of the Perrin sequence (right panel).
(The left two panels instead apply the Fibonacci numbers).

\(P(n)\) is also the solution of a third-order linear homogeneous recurrence equation having characteristic equation \(x^3-x-1=0\). The solutions to this equation are given by:$$\begin{align} x& = -\frac{1}{2} \, {\left(\frac{1}{18} \, \sqrt{23} \sqrt{3} + \frac{1}{2}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{-i \, \sqrt{3} + 1}{6 \, {\left(\frac{1}{18} \, \sqrt{23} \sqrt{3} + \frac{1}{2}\right)}^{\frac{1}{3}}}\\x &= -\frac{1}{2} \, {\left(\frac{1}{18} \, \sqrt{23} \sqrt{3} + \frac{1}{2}\right)}^{\frac{1}{3}} {\left(-i \, \sqrt{3} + 1\right)} - \frac{i \, \sqrt{3} + 1}{6 \, {\left(\frac{1}{18} \, \sqrt{23} \sqrt{3} + \frac{1}{2}\right)}^{\frac{1}{3}}}\\ x &= {\left(\frac{1}{18} \, \sqrt{23} \sqrt{3} + \frac{1}{2}\right)}^{\frac{1}{3}} + \frac{1}{3 \, {\left(\frac{1}{18} \, \sqrt{23} \sqrt{3} + \frac{1}{2}\right)}^{\frac{1}{3}}}\end{align} $$The third and only real-valued solution above is the so-called plastic constant \( \approx 1.32471795... \), also called the:

• le nombre radiant
• minimal Pisot number
• plastic number
• plastic ratio
• platin number
• Siegel's number
• silver number
• silver constant 

It is the limiting ratio of the successive terms of the Padovan sequence or Perrin sequence. Thus we have:$$\lim_{n \to \infty} \frac{P(n)}{P(n-1)}=P \approx 1.32471795$$The Padovan sequence mentioned is similar to the Perrin except that it has different starting values:$$P(n)=P(n-2)+P(n-3) \text{ where }P(0)=0,P(1)=1,P(2)=1$$It should be noted that for OEIS A050443, the ratio of successive terms does not approach the plastic constant but instead 1.22074... which as far as I can determine doesn't have a particular name.

Figure 3

It was only after composing this post that I realised I had already made a post about the plastic number on Friday, 5th June 2020. This is well worth the read because it illustrates some applications of the number (as shown in Figure 3). The Greek letter \( \rho \) can be used to represent the plastic ratio. In which case, to quote from the earlier post and referencing Figure 3:

There are precisely three ways of partitioning a square into three similar rectangles:

• The trivial solution given by three congruent rectangles with aspect ratio 3:1.  

• The solution in which two of the three rectangles are congruent with the third one of twice the linear dimension of the congruent pair and where the rectangles have aspect ratio 3:2.  

• The solution in which the three rectangles are mutually non-congruent (all of different sizes) and where they have aspect ratio \( \rho^2 \). 

The ratios of the linear sizes of the three rectangles are: 

• \( \rho \) (large : medium) 

• \( \rho^2\)(medium : small) 

• \( \rho^3 \) (large : small). 

The internal, long edge of the largest rectangle (the square's fault line) divides two of the square's four edges into two segments each that stand to one another in the ratio ρ. The internal, coincident short edge of the medium rectangle and long edge of the small rectangle divides one of the square's other, two edges into two segments that stand to one another in the ratio \(\rho^4\).

Figure 4 shows the approximate measurements for a unit square in which the three rectangles are mutually non-congruent. We have:$$ \frac{b}{1-b}=\rho$$

Figure 4

The Plastic Number

One of the delights of investigating the number of one's diurnal age, as I'm addicted to doing, is that every so often something fascinating pops up. Today, day number 25996, was one such day. The Online Encyclopedia of Integer Sequences (OEIS) didn't throw up much of interest. However, my newly discovered Sequence Database did reveal an interesting looking sequence, although one with an unprepossessing name: Sequence vbihjo41hob4e. It runs thus, up to 25996:

0, 3, 5, 2, 7, 6, 8, 12, 13, 19, 24, 31, 42, 54, 72, 95, 125, 166, 219, 290, 384, 508, 673, 891, 1180, 1563, 2070, 2742, 3632, 4811, 6373, 8442, 11183, 14814, 19624, 25996, ...

It arises from the recurrence relationship:$$a_n=a_{n-2}+a_{n-3}-1 \text{ where } a_0=0, a_1=3, a_2=5 \text { and } n \geq 3$$Of course, it's reminiscent of the Fibonacci sequence in that it adds two previous terms to get the next one, although it skips the immediately preceding term and also subtracts 1. I knew that the ratio of one term to the preceding term in the Fibonacci sequence approaches the Golden Ratio and so I investigated the ratio for this sequence. I found that:$$a_{101}:a_{100} \approx 1.32471795724460$$This intrigued me and I wondered if this number could be an approximation to some mathematical constant. Fortunately, I have a link to a website called Mathematical Constants and Sequences and it didn't take me long to discover what this constant was. See Figure 1 (click to enlarge).

Figure 1

As can be seen, 1.324717957244746025960908 ... is referred to as the plastic number \( \rho \) or silver constant. It's the solution to the polynomial equation \(x^3=x+1\) which has the following real solution:$$\sqrt[\leftroot{-2} \uproot{10} 3]{\frac{9 + \sqrt{69}}{18}} + \sqrt[\leftroot{-2} \uproot{10} 3]{\frac{9 - \sqrt{69}}{18}}$$The essential recurrence relationship is that of:$$a_n=a_{n-2}+a_{n-3} \text{ with } n \geq 3$$If we set \(a_0=a_1=a_2=1\), then we have the Padovan sequence: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... which is OEIS A000931.

My Sequence vbihjo41hob4e is essentially the Padovan sequence with different starting values \(a_0=0, a_1=3, a_2=5\) versus \(a_0=a_1=a_2=1\) and with 1 subtracted. The subtraction just means that it takes the numbers longer to get larger but addends or subtrahends and different starting values don't alter the long term behaviour. The ratio of one term to its predecessor always approaches the plastic number or silver constant. Subtracting integers greater than 1 leads to negative terms but the ratio still holds. Here is a permalink to SageMathCell that allows you experiment with the effect of different values.

The plastic number occurs in a variety of situations. Figure 2 shows one of them.

Figure 2: Three partitions of a square into similar rectangles

There are precisely three ways of partitioning a square into three similar rectangles:

The trivial solution given by three congruent rectangles with aspect ratio 3:1. The solution in which two of the three rectangles are congruent with the third one of twice the linear dimension of the congruent pair and where the rectangles have aspect ratio 3:2. The solution in which the three rectangles are mutually non congruent (all of different sizes) and where they have aspect ratio \( \rho^2 \). The ratios of the linear sizes of the three rectangles are: \( \rho \) (large : medium); \( \rho^2 \)(medium : small); and \( \rho^3 \) (large : small). The internal, long edge of the largest rectangle (the square's fault line) divides two of the square's four edges into two segments each that stand to one another in the ratio \( \rho \). The internal, coincident short edge of the medium rectangle and long edge of the small rectangle divides one of the square's other, two edges into two segments that stand to one another in the ratio \( \rho^4 \). Wikipedia

The  term "silver constant" should not be confused with the "silver ratio" of 1 + √2, which is one of the ratios from the family of metallic means. I posted about metallic means on Sunday, 3rd of March 2019. 

on January 22nd 2021

The changes were cosmetic and largely LaTeX-related, focused on improving the appearance of the cube root
symbol by raising the position of the 3 from its original default position using \leftroot {} and \uproot{}