Solving a Nonlinear Recursion

In the course of some research computations I have been doing, I run up against a recursion $$ a_{n+3} = a_{n+2}a_{n+1} - a_n $$

I've tried to find out if it's possible to solve recursions of this form, but can't find much since it's nonlinear. Does anyone know of methods that might be applicable; or failing that, if there are any assumptions on the initial conditions which might make it solvable?

Hope this is clear; I don't have any background in number theory or discrete math. Thanks.

Solution 1:

Let $F_1,F_2,\dots$ be a Fibonacci-like sequence, such that $F_{n}=F_{n-1} + F_{n-2}$, and let $$a_{n} = e^{F_{n}} + e^{-F_{n}} = 2\cosh F_n.$$ Then $$ \begin{eqnarray} a_{n-1}a_{n-2} &=& \left(e^{F_{n-1}} + e^{-F_{n-1}}\right) \left(e^{F_{n-2}} + e^{-F_{n-2}}\right) \\ &=& e^{F_{n-1} + F_{n-2}} + e^{F_{n-1} - F_{n-2}} + e^{-F_{n-1} + F_{n-2}} + e^{-F_{n-1} - F_{n-2}} \\ &=& e^{F_{n}} + e^{F_{n-3}} + e^{-F_{n-3}} + e^{-F_{n}} \\ &=& a_{n} + a_{n-3}, \end{eqnarray} $$ which is exactly your recursion. This family of solutions, parametrized by $(F_1, F_2)$, only covers part of the full range of initial conditions $(a_1, a_2, a_3)$. In particular, it will cover those cases where $$ a_3 = \frac{1}{2} a_1 a_2 \pm \frac{1}{2}\sqrt{\left(a_1^2-4\right)\left(a_2^2-4\right)}. $$