## The Fibonacci Sequence in Clojure

I was playing around this morning, and I decided to see what I could find about implementing the Fibonacci sequence in Clojure. For those unfamiliar with the Fibonacci sequence, which is named for a 13th-century Italian mathematician, it is a deceptively simple mathematical sequence where each term is constructed by the following rules:

• If n=0, then fib(n) = 0
• If n=1, then fib(n) = 1
• Otherwise, fib(n) = fib(n-1) + fib(n-2)

Traditionally, fib(n) is constrained to the set of positive integers.

This sequence doesn’t at first seem earth–shatteringly complex or interesting, but it turns out to describe all sorts of things in nature, such as the arrangement of leaves on stems, petals on flowers, and the family trees of bees. It also shows up in many other places in the fields of mathematics, computer science, and even music.

Implementing the Fibonacci sequence in various programming languages is a common exercise. A common solution uses recursion. A simple recursive solution to calculate the Fibonacci sequence in Clojure might look like this:

```(defn recursive-fibonacci
"Recursive implementation of the Fibonacci sequence."
[n]
(cond
(= n 0) 0
(= n 1) 1
:else (+ (recursive-fibonacci (- n 1))
(recursive-fibonacci (- n 2)))))

(map recursive-fibonacci [0 1 2 3 4 5 6 7 8 9])
```

You might implement something like this and think your job is done. But the recursive solution has two big problems: It’s verrrry slow, and if you give it a very big value of n, the Java Virtual Machine in which your Clojure code is running will eventually run out of heap space and crash. So let’s look at a couple other ways we could solve this problem in Clojure.

## Tail Recursion

Tail recursion allows the runtime to reuse memory locations for each subsequent call to our function. In Clojure, this is accomplished by the recur form, which might look like this:

```(defn tail-recursive-fibonacci
"Tail-recursive version of the fibonacci sequence."
[n]
(if (> n 1)
(loop [x 1
fib0 0
fib1 1]
(if (< x n)
(recur
(inc x)
fib1
(+ fib0 fib1))
fib1))
n))

(map tail-recursive-fibonacci [0 1 2 3 4 5 6 7 8 9])
```

What’s happening here? The recur form in Clojure implements tail recursion. So, let’s look at what happens if we invoke (tail-recursive-fibonacci 6):

• On entry, n=6 (so we do the looping thing instead of just returning n)
• Now we go through the loop:
• x = 1, fib0 = 0, fib1 = 1
• x = 2, fib0 = 1, fib1 = 1
• x = 3, fib0 = 1, fib1 = 2
• x = 4, fib0 = 2, fib1 = 3
• x = 5, fib0 = 3, fib1 = 5
• x = 6, fib0 = 5, fib1 = 8
• We’ve got our answer, (tail-recursive-fibonacci 6) = 8.

This is more memory-efficient and won’t crash if you give it a bigger number for n, but let’s look at a couple of more idiomatic ways to solve the problem.

## lazy-seq

A lazy sequence essentially creates an iterable sequence whose values are calculated as needed and then cached. This is more efficient than a recursive function, and also runs faster. We can create a lazy sequence using the lazy-seq form.

The lazy sequence version of the Fibonacci sequence looks like this:

```(def lazy-sequence-fibonacci
"Create a lazy sequence version of the Fibonacci sequence."
(
(fn fib [a b] (lazy-seq (cons a (fib b (+ a b)))))
0 1))

(take 10 lazy-sequence-fibonacci)
```

Something to notice here is that we’re using def instead of defn. This is because we’re not creating a function; rather, we’re creating a sequence that provides the next element each time we access it. (That’s also why I’ve switched my sample invocations from here out to use take instead of map - ClojureScript was giving me errors in the browser. I guess you can’t take something that’s not “seqable” in ClojureScript.)

## lazy-cat

Clojure also provides the lazy-cat form, which is a shortcut that essentially does a concat inside of a lazy-seq to collect our results. Here’s what the Fibonacci sequence looks like with lazy-cat:

```(def lazy-cat-fibonacci
(lazy-cat [0 1]
(map + (rest lazy-cat-fibonacci) lazy-cat-fibonacci)))

(take 10 lazy-cat-fibonacci)
```

## iterate

One more example. This one uses the iterate form. iterate takes a sequence, and on each iteration it returns the next value from the sequence and also a closure that will return the next element. It looks like this:

```(def seq-iterate-fibonacci
(map first (iterate (fn [[a b]] [b (+ a b)]) [0 1])))

(take 10 seq-iterate-fibonacci)
```

### Comparing Performance

To be honest, I’m new enough to Clojure that I’m not sure I totally understand the relative advantages and disadvantages of lazy-seq, lazy-cat and iterate, and why you’d use one versus the others. (I’m going to try to work it out, and then I’ll do another blog post!) But I did time the execution speed of these various methods, collecting the first 80 Fibonacci numbers from each. Here’s what I got running my test code in the REPL:

```Clojure 1.10.1
***********************************************************************
*          Fibonacci Sequence - Clojure Implementation Tester         *
***********************************************************************
=> recursive fibonacci - 50  repetitions:
"Elapsed time: 0.036135 msecs"

=> tail recursive fibonacci - 50  repetitions:
"Elapsed time: 0.004389 msecs"

=> lazy-seq fibonacci - 50  repetitions:
"Elapsed time: 0.004584 msecs"

=> lazy-cat fibonacci - 50  repetitions:
"Elapsed time: 0.003567 msecs"

=> iterate fibonacci - 50  repetitions:
"Elapsed time: 0.004653 msecs"

true
user=>
```

There wasn’t a lot of performance variation in my tests, except that the recursive implementation was consistently a LOT slower (913% slower, in this test run.)

Even if I don’t yet understand the reasons for picking between these various ways of solving the same problem, I do have a better understanding of the tools at my disposal.