# Recognizing a Factoring Pattern (Pt. 2)

I am trying to identify a pattern in the following set of equations;

$$N_{-1}=1$$

$$N_{0}=2y$$

$$N_{1}=2y^2+z$$

$$N_{2}=2y^3+3yz$$

$$N_{3}=2y^4+5y^2 z+z^2$$

$$N_{4}=2y^5+7y^3 z+4yz^2$$

$$N_{5}=2y^6+9y^4 z+9y^2 z^2+z^3$$

I need to express $$N_h$$. Any assistance would be greatly appreciated.

might be $$N_{j+2} = y N_{j+1} + z N_j$$
• This does seem to be the case, and if so, then the closed form (by the standard method for solving linear recurrence sequences) is $$N_k = \frac1{2^k} \left(y-\frac{y^2+z}{\sqrt{y^2+4 z}}\right) \left(y-\sqrt{y^2+4 z}\right)^k+\frac1{2^k} \left(y+\frac{y^2+z}{\sqrt{y^2+4 z}}\right) \left(y+\sqrt{y^2+4 z}\right)^k.$$ – Greg Martin May 22 at 5:10