Proof By Induction Fibonacci Numbers How do I prove that
$$ f_{ 2n+1 } = 3f_{ 2n } + 1 - f_{ 2n-3 } $$
I'm not sure how to prove it using the defining recurrence of Fibonacci numbers.

Let's do this the most basic way:
$f_{2n+1} = 3f_{2n} + 1 - f_{2n-3}$
$f_{2n} + f_{2n-1} = 3f_{2n} + 1 - f_{2n-3}$
$f_{2n-1} = 2f_{2n} + 1 - f_{2n-3}$
$f_{2n-1} + f_{2n-3} = 2f_{2n} + 1$
$f_{2n-1} + f_{2n-3} = f_{2n} + f_{2n-1} + f_{2n-2} + 1$
$f_{2n-3} = f_{2n} + f_{2n-2} + 1$
$f_{2n-3} = f_{2n} + f_{2n-3} + f_{2n-4} + 1$
$0 = f_{2n} + f_{2n-4} + 1$
Right side is strictly positive, which shows that the first equation didn't hold at first.
