The $(a,b,m)$-sequence, where $0 \leq a,b \lt m$, is defined as
$\begin{align*}
g(0)&=a\\
g(1)&=b\\
g(n)&= \big(g(n-1) + g(n-2)\big) \bmod m
\end{align*}$
All $(a,b,m)$-sequences are periodic with period denoted by $p(a,b,m)$.
The first few terms of the $(0,1,8)$-sequence are $(0,1,1,2,3,5,0,5,5,2,7,1,0,1,1,2,\ldots )$ and so $p(0,1,8)=12$.
Let $\displaystyle s(m)=\sum_{a=0}^{m-1}\sum_{b=0}^{m-1} p(a,b,m)^2$. For example, $s(3)=513$ and $s(10)=225820$.
Define $\displaystyle S(M)=\sum_{m=1}^{M}s(m)$. You are given, $S(3)=542$ and $S(10)=310897$.
Find $S(10^6)$. Give your answer modulo $999\,999\,893$.
Solution
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