# A Sum Of Products Of Sequences

$\begin{cases} x_{n}=(x_{n-1}+1)^{2}+2 \left \{ \frac{n}{2} \right \}\quad (n\ge 2) \\ x_{1}=4 .\end{cases}$ Define $$x_{n}$$ as an infinite sequence of numbers, as shown above.

Define $$y_{n}$$ as an infinite sequence of non-negative integers such that $$n|y_{n}$$ for all $$n\ge 1$$.

What is the largest integer that cannot be expressed in the form $\sum_{n=1}^{j}x_{n}y_{n}$ for some positive integer $$j$$?

Notations:

• $$n|y_{n}$$ means that $$n$$ divides $$y_{n}$$. Note that $$n|0$$ for $$n \neq 0$$.
• $$\{\cdot \}$$ denotes the fractional part function. For example, $$\{ 12.3345 \}=0.3345$$.
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