# Visualizing the Fibonacci sequence

Here are few interesting things I came across recently when I was thinking about the Fibonacci sequence. In particular, it involves visualizing the Fibonacci sequence.

The Fibonacci sequence is defined by the recurrence

f(n) =f(n– 1) +f(n– 2)

where

f(0) = 0

and

f(1) = 1

It's easy to show that we can also write

f(n) = (a–^{n}b) / (^{n}a–b)

where

a= (1 + √5) / 2

and

b= (1 - √5) / 2

It's also easy to see the behavior of *f*(*n*) for negative values of *n: *we have that

f(-n) = (-1)^{n}^{+1 }f(n)

What happens if we think of *f* as a function of a real variable instead of an integer variable?

In this case we have the function

f(x) = (a–^{x}b) / (^{x}a–b)

This turns out to be a complex-valued function that happens to have a real part of zero at integer values. If we graph *f*(*x*) for 0 ≤ *x ≤ *5, it looks like this (the complex value *z* = *x* + *i y* is plotted as (*x*, *y*)):

If we graph *f*(*x*) for -5 ≤ *x ≤ *0, it looks like this:

In that graph we can easily see how the fact *f*(-*n*) = (-1)^{n}^{+1 }*f*(*n*) is reflected.

To see what the graph for both positive and negative values of *x* look like together, here's the graph of *f*(*x*) for -4 ≤ *x ≤ 5:*

What about derivatives of *f*(*x*)?

It's fairly easy to see that

f^{(n)}(x) = (α_{n}a- β^{x}_{n}b^{x}) / (a–b)

where

α

= (log_{n}a)^{n}

and

β

= (log_{n}b)^{n}

so that the graph of *f*^{(n)} looks somewhat similar to the graph of *f*.

Here's how the graphs of *f*, *f*' and *f*'' look, for example:

If we look at how the differences between Fibonacci numbers behave, like

Δf(n) =f(n+ 1) –f(n) =f(n– 1)

we might expect to see this reflected in *f*', but there doesn't appear to be a nice, clean relationship like

f'(x) =f(x– 1)

I'm not even sure how close you can get in this particular case.

## Luke O'Connor

Hi Luther, nice post and diagrams. Did you make them with Maple? I had never seen the Fibonacci sequence plotted for negative values

rgs Luke

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## Luther Martin

Luke,

I actually used Mathematica. That’s probably my favorite piece of software of all time.

Luther

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