'''
It is possible to show that the square root of two can be expressed as an infinite continued fraction.

By expanding this for the first four iterations, we get:

The next three expansions are 
 
, 
 
, and 
 
, but the eighth expansion, 
 
, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.

In the first one-thousand expansions, how many fractions contain a numerator with more digits than the denominator?
'''

import math
import time

def t(n, dDict):
    assert n >= 0

    if n == 0:
        return 1, 1
    
    num, denom = d(n-1, dDict)
    return (num + denom, denom)

def d(n, dDict):
    assert n >= 0

    if n == 0:
        return 1, 2
    
    if n in dDict.keys():
        return dDict[n]
    
    num, denom = d(n-1, dDict)
    r = (denom, 2 * denom + num)
    dDict[n] = r
    return r

def main():
    print("Hello, this is Patrick")
    t0 = time.time()

    counter = 0
    dDict = {}
    for n in range(1001):
        num, denom = t(n, dDict)
        if len(str(num)) > len(str(denom)):
            counter += 1

    print(counter, time.time() - t0)

if __name__ == "__main__":
    main()