A Lucas Sum II

Calculus Level 4

Define the sequence \(\lbrace L_{n} \rbrace_{n=0}^{\infty}\) by \(L_{0} = 2\), \(L_{1} = 1\), and \(L_{n+2} = L_{n+1}+L_{n}\). If

\[\sum_{n=0}^{\infty} \frac{L_{n}^{2}}{4^{n}}\]

can be expressed in the form \(\frac{a}{b}\) for positive coprime integers \(a\) and \(b\), find \(a+b\).

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