Recurrence relations for $a_{n+2}$

Question

I'm trying to figure out how to find closed form equations for recurrence relations. I can find lots of examples for solving equations such as $a_{n} = ca_{n-1} + ca_{n-2}$ and $a_{n+1} = ca_{n} + ca_{n-1}$, but I came across a problem of $a_{n+2} = ca_{n+1} + ca_{n}$. How would the typical method for finding closed form equations (like the one described here) change for this problem?

Answer 1

All three equations you give are (essentially) the same. For example, put $n+2= m$ to reduce the third to the first.

Answer 2

Set $b_n=a_{n+2}$ or $d_n=a_{n+1}$ and solve for $b_n$ or $d_n$