Quadratic Residue of Fibonacci? Sounds Crazy!

Find the sum of all the Fibonacci numbers FnF_n less than 1 billion which follow the condition 13(4Fn222Fn+11)13 \mid (4 F_n^2 -22F_n+11)

Details and assumptions:

  • Fibonacci sequence is defined as F0=0,F1=1F_0=0,F_1=1 and for n2n\geq 2, Fn=Fn1+Fn2F_n=F_{n-1}+F_{n-2}. Thus, the Fibonacci sequence is 0,1,1,2,3,5,8,13,0,1,1,2,3,5,8,13,\ldots .

  • FnF_n denote the nn-th Fibonacci number.

This problem is a part of the set Crazy Fibonacci.
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