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); 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 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