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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