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





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

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

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

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


might be $$ N_{j+2} = y N_{j+1} + z N_j $$

Try it

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

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