The \(n^{\text{th}}\) **spread polynomial**, denoted by \(S_n(s)\), is recursively defined by \(S_0(s) \equiv 0\), \(S_1(s) \equiv s\), and

\( S_n(s) \equiv 2(1-2s) S_{n-1}(s) - S_{n-2}(s) + 2s, \)

for \(n \geq 2\). We will, for the context of this note, restrict our discussion to the finite field \(\mathbb{F}_p\), where \(p\) is an odd prime. As an example, we see that in \(\mathbb{F}_3\),

\( S_{0}(0) = 0 \quad s_{0}(1) = 0 \quad s_{0}(2) = 0 \)

\( S_{1}(0) = 0 \quad s_{1}(1) = 1 \quad s_{1}(2) = 2 \)

\( S_{2}(0) = 0 \quad s_{2}(1) = 0 \quad s_{2}(2) = 1 \)

\( S_{3}(0) = 0 \quad s_{3}(1) = 1 \quad s_{3}(2) = 0 \)

\( S_{4}(0) = 0 \quad s_{4}(1) = 0 \quad s_{4}(2) = 2 \)

\( S_{5}(0) = 0 \quad s_{5}(1) = 1 \quad s_{5}(2) = 1 \)

\( S_{6}(0) = 0 \quad s_{6}(1) = 0 \quad s_{6}(2) = 0 \)

In fact, for any positive integer \(n\) we will have that \(S_n(s) = S_{n+6}(s)\). Thus, a natural question to be asked is whether this phenomenon is also seen for arbitrary finite fields; as it turns out, it does and in fact we have \(S_n(s) = S_{n+k}(s)\) in \(\mathbb{F}_p\), where \( k \equiv \frac{1}{2}(p-1)(p+1) \). How do we prove that this is true? Are we able to discern other patterns within particular values of \(s\)? If so, how do we ascertain this mathematically?

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