# Recursive substitution

Algebra Level 5

$a_{n+2}-(2^{n+1}+2^{-n})a_{n+1}+a_n=0$

Consider the recurrence relation above. If $$a_1=1$$ and $$a_2=3$$, and$\lim_{n \to \infty} 2^{\frac{n(1-n)}{2}} a_n=\frac{1}{a} \left(1+\vartheta_b \left(c, \frac{1}{a}\right) \right)$what is the value of $$a^2+b^2+c^2$$?

Details and Assumptions:

$$\vartheta_z (x, y)$$ is the Jacobi theta function.

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