An integer formula for Fibonacci numbers
This code, somewhat surprisingly, generates Fibonacci numbers.
In this blog post, I’ll explain where it comes from and how it works.
Before getting to explaining, I’ll give a whirlwind background overview of Fibonacci numbers and how to compute them. If you’re already a maths whiz, you can skip most of the introduction, quickly skim the section “Generating functions” and then read “An integer formula”.
The Fibonacci numbers are a well-known sequence of numbers:
The th number in the sequence is defined to be the sum of the previous two, or formally by this recurrence relation:
I’ve chosen to start the sequence at index 0 rather than the more usual 1.
Computing Fibonacci numbers
There’s a few different reasonably well-known ways of computing the sequence. The obvious recursive implementation is slow:
An iterative implementation works in operations:
And a slightly less well-known matrix power implementation works in operations.
The last method works by considering the
as sequences, and noting that
From which follows
and so if then (noting that unlike Python, matrix indexes are usually 1-based).
It’s based on the assumption that numpy’s matrix power does something like exponentation by squaring.
Another method is to find a closed form for the solution of the recurrence relation. This leads to the real-valued formula: where and . The practical flaw in this method is that it requires arbitrary precision real-valued arithmetic, but it works for small .
A generating function for an arbitrary sequence is the infinite sum . In the specific case of the Fibonacci numbers, that means . In words, it’s an infinite power series, with the coefficient of being the th Fibonacci number.
Multiplying by and summing over all , we get:
If we let to be the generating function of , which is defined to be then this equation can be simplified:
We can solve this equation for to get
It’s surprising that we’ve managed to find a small and simple formula which captures all of the Fibonacci numbers, but it’s not yet obvious how we can use it. We’ll get to that in the next section.
A technical aside is that we’re going to want to evaluate at some values of , and we’d like the power series to converge. We know the Fibonacci numbers grow like and that geometric series converge if , so we know that if then the power series converges.
An integer formula
Now we’re ready to start understanding the Python code.
To get the intuition behind the formula, we’ll evaluate the generating function at .
Interestingly, we can see some Fibonacci numbers in this decimal expansion: . That seems magical and surprising, but it’s because .
In this example, the Fibonacci numbers are spaced out at multiples of , which means once they start getting bigger that 1000 they’ll start interfering with their neighbours. We can see that starting at 988 in the computation of above: the correct Fibonacci number is 987, but there’s a 1 overflowed from the next number in the sequence causing an off-by-one error. This breaks the pattern from then on.
But, for any value of , we can arrange for the negative power of 10 to be large enough that overflows don’t disturb the th Fibonacci. For now, we’ll just assume that there’s some which makes sufficient, and we’ll come back to picking it later.
Also, since we’d like to use integer maths (because it’s easier to code), let’s multiply by , which also puts the th Fibonacci number just to the left of the decimal point, and simplify the equation.
If we take this result modulo , we’ll get the th Fibonacci number (again, assuming we’ve picked large enough).
Before proceeding, let’s switch to base 2 rather than base 10, which changes nothing but will make it easier to program.
Now all that’s left is to pick a value of large enough so that . We know that the Fibonacci numbers grow like , and , so is safe.
So! Putting that together:
If we use left-shift notation that’s available in python, where then we can write this as:
Observing that can be expressed as the bitwise and (
&) of , we reconstruct our original Python program:
Although it’s curious to find a non-iterative, closed-form solution, this isn’t a practical method at all. We’re doing integer arithmetic with integers of size bits, and in fact, before performing the final bit-wise and, we’ve got an integer that is the first Fibonacci numbers concatenated together!