Non-Fibonacci numbers, appear from Fibonacci Sums ...?

Question

(Please consider that i am not the best on explaining things when it comes to mathematics, so i will try my best.)

Lately this month i had an idea, to test if the Sum of Fibonacci Numbers always gives a specific pattern. Testing it programmatically i found out that:

If we have for example a List of Fibonacci Numbers, Fib[n] = 0,1,1,2,3,5,8,... and try to Sum them step by step (i by i) and insert each sum, in another list named (for example) S_Fib[n], like below:

[Programmatically]

 Dim Sum As BigInteger
 Dim  n  As Integer

 For i = 0 To n             '[Loop from 0 to n]

     Sum = 0                '[Sum value goes to 0]
     For j = 0 To i         '[Sums Fib. Numbers from 0 to i]
         Sum = Sum + Fib(j)
     Next

 S_Fib.Add(sum)             '[Inserting The sum from 0 to i 
                            ' of Fib. Numbers, into the S_Fib
                            ' i(th) position]
 Next

[Mathematically] (i am really sorry if i have anything too wrong)

$$S\_Fib[n] =\sum^n_{i=0}{(Fib[i])} $$

We are getting a patern like this:

Fib  [n] = {0, 1, 1, 2, 3, 5 , 8 , 13, 21, 34, 55 , 89 , 144, 233, 377,..} 
S_Fib[n] = {0, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986,..}

Where after the number for n=2, none of the other numbers in S_Fib list is Fibonacci and also it appears that the S_Fib[n] = Fib[n+2] - 1 (where n is a pointer in the list)

just in case/to be sure, that i didn't up everything while i was trying to explain, here is an example:

n = 3

Fib [n+2] = 5
S_Fib [n] = 0 + 1 + 1 + 2 = 4

S_Fib[n] = Fib[n+2] - 1 => 4 = 5 - 1 <=> 4 = 4

So, my question is, Why? what is the proof of this "concept" Mathematically if it is actually valid at all...and is there any site or whatever, where i can check about it?

Thanks in advance for any reply and i am really sorry if my question was indeed too "dumb" :P to ask for [...]