Tweaking the Fibonacci Sequence!

Calculus Level 5

Let \(k \in (0,1)\), and let the sequence \((B_n)_{n=0}^{\infty}\) be defined by:

\(B_0 = k, B_1 = k^2\) and \(B_{n+2} = kB_{n+1} + k^2B_n\) for integers \(n \geq 0\).

If the value of the expression \(\large{\displaystyle \sum_{n=0}^{\infty} \dfrac{B_n}{n+1}}\) can be expressed as:

\[\large{\dfrac1{\sqrt{\gamma}} \ln \left(\dfrac{1-\beta k}{1-\alpha k}\right)}\]

where \(\gamma \in \mathbb Z; \alpha, \beta \in \mathbb R\) and \(\alpha >0, \beta<0\). Find the value of \(\large{\lfloor \alpha + \beta + \gamma \rfloor}\).


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