I have been experimenting with generalizing the Fibonacci sequence, and Fibonacci-like "metallic mean" sequences such as the Pell sequence, to non-integer and complex values.
The standard, real-integer form of a metallic mean sequence is
$$M(a, n) =
\begin{cases}
0 & \text{if $n = 0$} \\
1 & \text{if $n = 1$} \\
a × M(a, n-1) + M(a, n-2) & \text{otherwise.}
\end{cases}$$
When $a = 1$ this gives the Fibonacci (golden) sequence (0, 1, 1, 2, 3, 5, …), when $a = 2$ this gives the Pell (silver) sequence (0, 1, 2, 5, 12, 29, …), when $a = 3$ this gives the bronze sequence (0, 1, 3, 10, 33, 109, …), and so on.
This already works for non-integer, complex values of $a$, but to generalize this to non-integer values of $n$ we need a different formula.
Ratios of sequential Fibonacci numbers approach the golden ratio $δ_{g+} = \frac{1 + \sqrt{5}}{2}$. The Fibonacci numbers grow as a function of the golden ratio $δ_{g+}$ and the golden ratio conjugate $δ_{g-} = 1 – δ_{g+} = -\frac{1}{δ_{g+}} = \frac{1 – \sqrt{5}}{2}$:
$$F(n) = \frac{δ_{g+}^n – δ_{g-}^n}{δ_{g+} – δ_{g-}}$$
Ratios of sequential Pell numbers approach the silver ratio $δ_{s+} = \frac{2 + \sqrt{8}}{2}$. The Pell numbers grow as a function of the silver ratio $δ_{s+}$ and the silver ratio conjugate $δ_{s-} = 2 – δ_{s+} = -\frac{1}{δ_{s+}} = \frac{2 – \sqrt{8}}{2}$:
$$P(n) = \frac{δ_{s+}^n – δ_{s-}^n}{δ_{s+} – δ_{s-}}$$
We can generalize the golden ratio and the silver ratio to the metallic mean $δ_+(a) = \frac{a + \sqrt{a^2 + 4}}{2}$, and their conjugates to the metallic mean conjugate $δ_-(a) = \frac{a – \sqrt{a^2 + 4}}{2}$. Then the full equation for non-integer values of $n$ is:
$$M(a, n) = \frac{δ_+(a)^n – δ_-(a)^n}{δ_+(a) – δ_-(a)} = \frac{δ_+(a)^n – δ_-(a)^n}{\sqrt{a^2 + 4}}$$
This general form allows for non-integer, complex values of both $a$ and $n$.
There are other equations that give the Fibonacci numbers:
- $Fe(n) = \frac{δ_{g+}^n – δ_{g+}^{-n}}{δ_{g+} – δ_{g-}}$ gives the even Fibonacci numbers for even integer values of $n$
- $Fo(n) = \frac{δ_{g+}^n + δ_{g+}^{-n}}{δ_{g+} – δ_{g-}}$ gives the odd Fibonacci numbers for odd integer values of $n$
- $Fa(n) = \frac{δ_{g+}}{δ_{g+} – δ_{g-}}$ rounded to the nearest integer gives all Fibonacci numbers for integer values of $n$
We can rewrite these equations using the metallic mean equations given earlier to get
- $Me(a, n) = \frac{δ_+(a)^n – δ_+(a)^{-n}}{\sqrt{a^2 + 4}}$
- $Mo(a, n) = \frac{δ_+(a)^n + δ_+(a)^{-n}}{\sqrt{a^2 + 4}}$
- $Ma(a, n) = \frac{δ_+(a)}{\sqrt{a^2 + 4}}$
If we create a 3D plot with real values of $n$ on the $x$-axis, the real component of the function output on the $y$-axis, and the imaginary component of the function output on the $z$-axis, and take $a$ to be some complex number while restricting $n$ to real numbers, the plot of $M(a, n)$ forms a spiral with the plot of $Ma(a, n)$ running down the spiral's center. $Me(a, n)$ curves along one side of the spiral, intersecting it at even integer values of $n$, while $Mo(a, n)$ runs along the opposite side of the spiral, intersecting it at odd integer values of $n$.
When looking at this 3D plot for various real and complex values of $a$, I noticed that as long as $a$ is entirely real, the curves given by the parametric equations
$$CurveMh_1(a, n) =
\begin{cases}
x = n \\
y = \text{Re}(Ma(a, n)) \\
z = -\text{Re}(Ma(a, -n))
\end{cases}$$
and
$$CurveMh_2(a, n) =
\begin{cases}
x = n \\
y = \text{Re}(Ma(a, n)) \\
z = \text{Re}(Ma(a, -n))
\end{cases}$$
where
$$\text{Re}(b + c i) = b$$
intersect the spiral of $M(a, n)$ at every half-integer ($\frac{5}{2}$, $\frac{3}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$, $-\frac{3}{2}$, and so on). However, when $a$ has a nonzero imaginary component, these curves no longer follow the half-integer points. This is in contrast to the functions $Ma(a, n)$, $Me(a, n)$, and $Mo(a, n)$, which continue to follow the spiral faithfully. Which finally leads to my question:
Are there equations of the form $f(a, n)$ whose outputs match the corresponding value of $M(a, n)$ for half-integer values of $n$ and complex values of $a$?
Best Answer
It turns out all I needed was to realize that the "+" and "−" in the $Me(a, n)$ and $Mo(a, n)$ equations was giving the phase of the resulting curve - that is, its rotation along the main spiral, with a phase of "+1" corresponding to odd integers along the "top" of the spiral and "−1" corresponding to even integers along the "bottom" of the spiral, 180° offset. As the spiral arises from continuous exponential powers of $i$ rotating through the values 1, $i$, −1, and −$i$, all I needed to do was replace the phases of "+1" and "−1" with "+$i$" and "−$i$" to get the values halfway in between. This gives the following expressions in answer to my stated question:
$$Mh_+(a, n) = \frac{δ_+(a)^x + ί δ_+(a)^{-x}}{\sqrt{a² + 4}}$$ $$Mh_-(a, n) = \frac{δ_+(a)^x - ί δ_+(a)^{-x}}{\sqrt{a² + 4}}$$
Additionally, an equation for any rotational angle along the spiral, corresponding to any point between integers, can be found by multiplying the second term in the numerator by a complex number with the desired argument and a magnitude of one.