Quadratic Residue of Fibonacci? Sounds Crazy!

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

Details and assumptions:

  • Fibonacci sequence is defined as \(F_0=0,F_1=1\) and for \(n\geq 2\), \(F_n=F_{n-1}+F_{n-2}\). Thus, the Fibonacci sequence is \(0,1,1,2,3,5,8,13,\ldots \).

  • \(F_n\) denote the \(n\)-th Fibonacci number.

This problem is a part of the set Crazy Fibonacci.

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