The plastic number, and Padovan and Perrin-like sequences

Question

The ratio of $P_{n+1}$ to $P_n$ tends toward the plastic number (https://en.wikipedia.org/wiki/Plastic_number) as $n$ approaches infinity for the Padovan and Perrin sequences. There are other sequences that have this property? What makes the Padovan and Perrin sequences the most fundamental of such sequences? Are they equally fundamental, whatever equally fundamental may mean in this case? What is the most fundamental way of describing the relation between these sequences?

Answer 1

Just about every sequence $(a_n)_{n \geq 0}$ that satisfies a linear recurrence $a_{n+3} = a_{n+1} + a_n$ has the property that $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho$, where $\rho$ is the unique real solution of the equation $x^3 = x + 1$. This is stated in the Wikipedia article that the OP links to.

Moreover, this is not hard to see; one way of seeing it is via generating functions. Form the power series $f(x) = \sum_{n \geq 0} a_n x^n$. Then

$$\begin{array}{lll} f(x) & = & a_0 + a_1 x + a_2 x^2 + \sum_{n \geq 0} a_{n+3} x^{n+3} \\ & = & a_0 + a_1 x + a_2 x^2 + \sum_{n \geq 0} (a_n + a_{n+1}) x^{n+3} \\ & = & a_0 + a_1 x + a_2 x^2 + x^3 f(x) + x^2 (f(x) - a_0) \end{array}$$

and so after a little rearrangement we get

$$f(x) = \frac{a_0 + a_1 x + (a_2 - a_0)x^2}{1 - x^2 - x^3}$$

By the method of partial fractions, such a rational function may rewritten in the form

$$f(x) = \frac{A_1}{1 - \rho_1 x} + \frac{A_2}{1 - \rho_2 x} + \frac{A_3}{1 - \rho_3 x}$$

where $A_1, A_2, A_3$ are constants whose exact expression we don't need to bother with; more important is the factorization $1 - x^2 - x^3 = (1 - \rho_1 x)(1 - \rho_2 x)(1 - \rho_3 x)$ where, dividing through by $x^3$ and putting $t = 1/x$, we find $t^3 - t - 1 = (t - \rho_1)(t - \rho_2)(t - \rho_3)$. So one of the roots, say $\rho_1$, is the plastic number $\rho$, and the other two are conjugate complex numbers (of absolute value less than $1$), also mentioned in the Wikipedia article.

Anyway, the point of this calculation is that the rational function $f(x)$ can now be re-expressed as a linear combination of geometric series

$$f(x) = A_1 \sum_{n \geq 0} \rho^n x^n + A_2 \sum_{n \geq 0} \rho_2^n x^n + A_3 \sum_{n \geq 0} \rho_3^n x^n$$

which by matching coefficients, gives us a reasonably explicit expression for $a_n$:

$$a_n = A_1 \rho^n + A_2 \rho_2^n + A_3 \rho_3^n$$

Because $|\rho_2| = |\rho_3| < 1$ and $|\rho| > 1$, it is clear that as $n$ approaches infinity, the dominant term is $A_1 \rho^n$ (provided that $A_1$ doesn't vanish!); the other terms converge to $0$. And so asymptotically, $a_n \approx A_1 \rho^n$. Similarly,

$$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{A_1 \rho^{n+1}}{A_1 \rho^n} = \rho$$

by an elementary manipulation, and this proves the claim.

As for the other questions, I have no way of evaluating which of various sequences satisfying this recurrence is to be considered "most fundamental", except perhaps on grounds of "simplicity". For example, one of those sequences (the Padovan sequence, is it?) gives the reasonably simple-looking $f(x) = \frac{1 + x}{1 - x^2 - x^3}$. But what the analysis above shows is that all such sequences are (complex) linear combinations of the three sequences $\rho^n$, $\rho_2^n$, $\rho_3^n$ -- and so perhaps those should be considered the most fundamental (again, on grounds of simplicity).

Answer 2

If I understand correctly, you are interested in the significance of the Padovan and Perrin sequences and how they are related to each other.

A distinguishing feature of the Padovan sequence is that it is a whorled triangle that grows as this sequence that will form a complete, closed mosaic that covers the plane, much as the Fibonacci mosaic is a whorled square. Moreover, the spiral that is composed of circular arcs that circumscribe the triangles, approximates a logarithmic spiral with a flair coefficient of $b=3\text{ln}p/\pi$, where $p$ is the plastic constant. Again, there is a parallel between the Fibonacci and Padovan sequences. The figure below shows the mosaic and Padovan spiral.

The Perrin number apparently has some significance in graph theory, but that is beyond my ken. You can get started on that here: Perrin Number.

Now, as pointed out elsewhere on this page, almost every sequence $P(n≥0)$ that satisfies a linear recurrence $P_{n+3}=P_{n+1}+P_n$ has the property that $$\lim_{n→∞}\frac{P_{n+1}}{P_n}=p$$

All such sequences can be found analytically from the Binet-like formula $$P(n)=ap^n+bq^n+cr^n$$ where $p, q \text{ and } r$ are the roots of the equation $$x^3=x+1$$ and the constants $a, b \text{ and } c$ are to be determine from the initial conditions, i.e., $[P(0),P(1),P(2)]$. Thus, $a, b \text{ and } c$ can be found from the linear equations

$$ \left\{ \begin{array}{c} a+b+c=P(0) \\ ap+bq+cr=P(1) \\ ap^2+bq^2+cr^2=P(2) \end{array} \right. $$

with the solution

$$\begin{array}{l}a = \frac{{ - P\left( 2 \right) + P\left( 1 \right)\left( {q + r} \right) - P\left( 0 \right)qr}}{{\left( {p - q} \right)\left( {p - r} \right)}}\\b = \frac{{P\left( 1 \right)p - P\left( 2 \right) - P\left( 0 \right)pr + P\left( 1 \right)r}}{{\left( { - p + q} \right)\left( {q - r} \right)}}\\c = {b^*}\end{array}$$

Now, since $b$ and $c$ are complex conjugates, as are $q$ and $r$, we can write

$$P(n)=ap^n+2\mathfrak{Re}\{bq^n\}$$

which shows that $P(n)$ is always real. As a side note, I have shown (as probably have many others) that $P(n)=\text{Round}(ap^n)$ for the Padovan sequence, at least. (I haven't really tried it on any others.)

So this little discussion shows how the Padovan and Perrin sequences are related. It's also worth noting that for the Perrin sequence $a=b=c=1$. And now the Padovan mosaic and spiral...