sum of even-valued and odd-valued Fibonacci numbers

Question

I was solving the Project Euler problem 2

*By starting with 1 and 2, the first 10 terms of Fibonacci Series will be:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Find the sum of all the even-valued terms in the sequence which do not exceed 4 million.*

here is my code in Python

a=0
b=1
nums = []
while True:
    x=a+b
    a=b
    b=x
    if(b>4000000):
        break
    nums.append(x)
sum=0
for x in nums:
    if(x%2==0):
        sum+=x
print sum

I noticed that the answer comes out to be 4613732 However I initially did a mistake by doing x%2!=0 and the answer turned out to be 4613731 (4613732-1) Is this some property or just luck??

Answer 1

Each even valued term (which are the 2nd, 5th, and every 3rd one after) is the sum of the preceding two odd terms. There is a startup transient that 2 doesn't have two 1's before it (you would if you started with 1,1). So the sum of the even valued terms up to some value will be 1 more than the sum of the preceding odd terms.

Answer 2

The sequence of Fibonacci numbers mod 2 goes $odd_1, odd_2, even_3$ - and in this sequence $odd_1+odd_2=even_3$. So the difference between the total of odds and evens up to any even number in the sequence is a constant, which happens to be 1. Stopping at an odd number gives a different result.