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(\$[^\n$]*[^\s$])(-|=|\+)([^\s$][^\n$]*\$)
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\end{theorem} A $k$-periodic sequence has the property that $s_i = s_{i + k}$ for all $i = 0,1,\dots$. Thus a $k$-periodic sequence $(s_i)_{i = 0}^\infty$ may be represented by any finite sequence $(s_i)_{i=a}^{a+k - 1}$, where $a$ is usually chosen to be $0$. Sadly our Fibonacci sequence examples are not defined over a finite field but over the naturals and thus are not necessarily periodic. Examples such as these may be interpreted to have a period of $\infty$. The period and related stability of linear recurrence sequences in regard to linear complexity has a very rich and broadly studied background~\cite{DingZiaoShan1991}. \begin{theorem} \label{th: max period is m-sequence} \cite[Theorem~6.33]{LidlNiederreiter1994} A linear recurrence sequence $s$ over a finite field $\gf_2$ with linear complexity $n$ has a maximum possible period of $2^n-1$. \end{theorem} \begin{definition} \label{de: m-sequence} A sequence which has maximum period for giv