function fib(n)
       if n = 0 or n = 1
           return 1
       return fib(n − 1) + fib（n − 2）

fib(5)
fib(4) + fib(3)
(fib(3) + fib(2)) + (fib(2) + fib(1))
((fib(2) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))
(((fib(1) + fib(0)) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))

>>> def fib_to(n):
...     fibs = [0, 1]
...     for i in range(2, n+1):
...         fibs.append(fibs[-1] + fibs[-2])
...     return fibs
...

>>> fib_to(20)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765]

>>> fib_to(40)[17]
1597

>>> def fib(n, computed = {0: 0, 1: 1}):
...     if n not in computed:
...         computed[n] = fib(n-1, computed) + fib(n-2, computed)
...     return computed[n]

>>> fib(400)
176023680645013966468226945392411250770384383304492191886725992896575345044216019675

import functools

@functools.lru_cache(None)
def fib(n):
    if n < 2:
        return n
    return fib(n-1) + fib(n-2)

>>> def fib(n):
...     a, b = 0, 1
...     for _ in range(n):
...         a, b = b, a+b
...     return a

cache = {}

def fib(n):
    if n <= 1:
        return n
    if n not in cache:
        cache[n] = fib(n-1) + fib(n-2)
    return cache[n]