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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