How can one convert the number sequence with $a_0 = 1$, $a_1 = 2$ and $a_n = 2a_{n - 1} + 3a_{n - 2}$ for $n\ge 2$ into its explicit form?

Question

How can one convert the number sequence with $a_0 = 1$, $a_1 = 2$ and $a_n = 2a_{n - 1} + 3a_{n - 2}$ for $n\ge 2$ into its explicit form?

Thanks in advance!

Answer 1

Use the Method of characteristic polynomials in order to solve this linear recurrence relation. Take $a_n=k\cdot r^n$. Then, the given recurrence relation implies $$k\cdot r^n=2\cdot k\cdot r^{n-1}+3\cdot k\cdot r^{n-2}\iff r^2=2r+3\iff (r+1)(r-3)=0$$ Therefore $$a_n=k_1\cdot (-1)^n+k_2\cdot 3^n$$ Let $n=0$ and $n=1$ in order to obtain $$a_n=\frac14\cdot \left((-1)^n+3^{n+1}\right)$$ Use induction to prove the result.