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Range of periodic sequence

 Published on Sunday, 11th October 2020, 05:00 am; Solved by 144;
Difficulty rating: 60%

Problem 729

Consider the sequence of real numbers $a_n$ defined by the starting value $a_0$ and the recurrence $\displaystyle a_{n+1}=a_n-\frac 1 {a_n}$ for any $n \ge 0$.

For some starting values $a_0$ the sequence will be periodic. For example, $a_0=\sqrt{\frac 1 2}$ yields the sequence: $\sqrt{\frac 1 2},-\sqrt{\frac 1 2},\sqrt{\frac 1 2}, \dots$

We are interested in the range of such a periodic sequence which is the difference between the maximum and minimum of the sequence. For example, the range of the sequence above would be $\sqrt{\frac 1 2}-(-\sqrt{\frac 1 2})=\sqrt{ 2}$.

Let $S(P)$ be the sum of the ranges of all such periodic sequences with a period not exceeding $P$.
For example, $S(2)=2\sqrt{2} \approx 2.8284$, being the sum of the ranges of the two sequences starting with $a_0=\sqrt{\frac 1 2}$ and $a_0=-\sqrt{\frac 1 2}$.
You are given $S(3) \approx 14.6461$ and $S(5) \approx 124.1056$.

Find $S(25)$, rounded to 4 decimal places.