How to find a closed formula for partial sums of recursively defined series with $t_{n} = t_{n-2} + t_{n-3}$?

Question

If $1,1,2,2,3,4,5,7,9,12,16,21,28,37,\ldots,n$ - terms, $t_{n} = t_{n-2} + t_{n-3}$. Find the sum of such a series up to $n$ terms

Progress

Attempted to solve the recurrence relation $t_{n} = t_{n-2} + t_{n-3}$, unsuccessfully.

Answer 1

You have a linear homogeneous recurrence relations with constant coefficients. You assume a solution of the form $cr^n$ and plug it in to get the characteristic polynomial $r^3=r+1$. As it is a cubic, there will be three roots (in this case two are complex). The solution is then $t_n=c_1r_1^n+c_2r_2^n+c_3r_3^n$. You need to evaluate the $c$'s from your initial conditions. Your polynomial doesn't have nice roots so this will be a mess.