# A Very Average Bunch Of Numbers

Algebra Level 5

Let $$x_0, x_1, x_2,\ldots$$ be a sequence of a numbers satisfying the recursion,

$x_n =\dfrac{x_{n-1} + x_{n+1}}2 ,$

for $$n = 1,2,3,\ldots$$.

Given that $$x_{343434}$$ and $$x_{515151}$$ are non-negative real numbers whose sum is $$5$$, find the number of possible integral values of $$x_0$$.

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