$n +1$th Fibonacci number modulo $n$

Question

The Pisano period studies the $n$th Fibonacci number $F_{n}$ modulo $n$. Is there anything about $F_{n + 1} \pmod n$?

Answer 1

The sequence $F_{n + 1} \pmod n$ is interesting enough to have been added to the Online Encyclopedia of Integer Sequences, but not interesting enough for its entry to have been expanded past the bare minimum: https://oeis.org/A002726

However, its plot does not indicate any periodicity: (log-y) (linear)

The plot indicates higher density of points around $y=1$, $y={x\over2}$, $y={x\over3}$ and $y={2\over3} x$, but the function looks random except for that. The positions of zeroes and ones do not indicate much regularity either - except all zeroes seem to be at prime positions.

Note, however, the sequence $F_n \pmod n$ that you have mentioned does not show any striking regularity either, except the dense lines are now just two, $y=1$ and $y=x$: https://oeis.org/A002708 (plot)

---

The pisano period is defined as the period of the sequence $F_n \pmod m$ - for any fixed $m$ this sequence is periodic with the period being a function of $m$. Stated mathematically, $F_n \equiv F_{n+\pi(m)} \pmod m$ where $\pi(m)$ is the Pisano sequence.

Ref:
http://en.wikipedia.org/wiki/Pisano_period
https://oeis.org/A001175

Answer 2

If you are looking for some regularity in the sequence, you have that for $p\ne 5$ prime: $$ F_{p+1} \operatorname{mod}p = \begin{cases} 1 \quad &\text{if the congruence } x^2 \equiv 5 \pmod{p}\text{ has a solution}\\0 \quad &\text{otherwise}\end{cases}$$