I know how to solve linear homogeneous and non-homogeneous recurrence relations. But today I found this problem.
$$F_1=1$$ $$F_2=7$$ $$F_3=13$$ $$F_n = 3F_{n-1} + k(F_{n-2}F_{n-3})+10$$
Where $k$ is a given number. A friend of mine had this problem on a programming homework. And I wanted to impress him and told him there is a formula for this problem. But when I looked it more closely I realized it is not like other such problems that I have solved in discrete mathematics course, where we solved linear recurrence relations. So my question is could I find a formula for this recurrence relation and if I can how?