Exact Frequencies of Consecutive Quadratic Residue and Nonresidue Patterns Modulo a Prime

This paper determines the exact frequency of all consecutive sign patterns of lengths two and three formed by the quadratic character modulo an odd prime $p$. Using basic properties of character sums over finite fields, we derive exact counting formulas for pairs $(n, n+1)$ and triples $(n-1, n, n+1)$ exhibiting specific sequence patterns in $\{\pm 1\}$. The frequencies of pairs are classified entirely by $p \pmod 4$, whereas triples exhibit a more complex dependency on $p \pmod 8$ and the Jacobsthal sum $T_p=\sum_{x\in\mathbb F_p}\chi(x^3-x)$.