Triples and quadruples of consecutive squares or non-squares in a finite field

Let $\text{F}$ be the finite field of odd prime power order $q$, We find explicit expressions for the number of triples $\{\text{al}-1,\text{al},\text{al}+1\}$ of consecutive non-zero squares in $\text{F}$ and similarly for the number of triples of consecutive non-square elements. A key ingredient is the evaluation of Jacobsthal sums over general finite fields by Katre and Rajwade. This extends results of Monzingo(1985) to non-prime fields. Curiously, the same machinery alows the evaluation of the number of consecutive quadruples $\{\text{al}-1,\text{al},\text{al}+1,\text{al}+2\}$ of square and non-squares over $\text{F}$, when $q$ is a power of 5.