# Generation phi

Let's define a series $$S$$, with $$0^{th}$$ element $${ a }_{ 0 }$$ as:

$S({ a }_{ 0 })={ \quad \{ a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 },{ a }_{ 4 },...\}$

Where $${ a }_{ n }=\quad \varphi { a }_{ n-1 }+\frac { 1 }{ \varphi } \quad or\quad { a }_{ n }=\frac { { a }_{ n-1 } }{ \varphi } -\frac { 1 }{ { \varphi }^{ 2 } }$$ with equal probability.

If $${ a }_{ 0 }=31$$, the expected value of $${a}_{51}$$ can be expressed as:

$\frac { { a }^{ b }\sqrt { c } }{ { d }^{ e } } -f$

Find: $$a+b+c+d+e+f$$

Details and Assumptions:

1. $$\varphi=\frac { \sqrt { 5 } +1 }{ 2 }$$
2. $$a,b,c,d,e,f$$ are integers. $$a$$ and $$d$$ are co-prime. $$c$$ is a square-free number.
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