In clojure, how to build lazy sequence using iterate function

Question

The clojure document gives the following examples:

(take 10 (iterate (partial + 2) 0))

(def powers-of-two (iterate (partial * 2) 1))
(take 10 powers-of-two)

(def fib (map first (iterate (fn [[a b]] [b (+ a b)]) [1 1])))
(take 10 fib)

Anyone can explain the syntax of clojure's iterate function in more detail? I am very confused with all the usage. Why two brackets are there in (fn [[a b]] [b (+ a b)])?

Another example can be found here:

(defn iter [[x y]]
  (vector y (+ x y)))

(nth (iterate iter [0 1]) 10000)

Answer 1

iterate takes a function f and an initial value x and produces a lazy sequence. The first element in the seq is x. Each subsequent element is computed by calling f with the previous element.

Example 1:

(iterate (partial + 2) 0)

This generates a sequence, starting at 0, where each element is the previous element with 2 added to it. I.e.:

0
(+ 2 0) ; => 2
(+ 2 2) ; => 4
(+ 2 4) ; => 6
; etc

Each element in the seq is passed to (partial + 2) when generating the following element.

Example 2:

(iterate (partial * 2) 1)

This generates a sequence, starting at 1, where each element is the previous element multiplied by 2. I.e.:

1
(* 2 1) ; => 2
(* 2 2) ; => 4
(* 2 4) ; => 8
(* 2 8) ; => 16
; etc

Again, you can see how each element feeds into the generation of the next one.

Example 3:

(iterate (fn [[a b]] [b (+ a b)]) [1 1])

Firstly, (fn [[a b]] ...) is a way to destructure a value into parts. In this case, the function accepts a two-element vector and unpacks it into the local variables a and b.

The function returns a two-element vector containing b and the sum of a and b (i.e. the second value in the previous pair and the sum of both values in the previous pair).

With this in mind, this iterate call generates:

[1 1]
[1 (+ 1 1)] ; => [1 2]
[2 (+ 1 2)] ; => [2 3]
[3 (+ 2 3)] ; => [3 5]
[5 (+ 3 5)] ; => [5 8]
; etc

Then (map first ...) grabs the first value in each pair, which gives you your Fibonacci sequence.