The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field ${\mathbb{F}}_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge 3$. In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements $a\in {\mathbb{F}}_{q^n}$ for which Trace$(a)=0$ and Trace$(a^{-1})=0$. Besides, we apply the formula to give an upper bound on the number of distinct constructions of a family of sequences with good family complexity and cross-correlation measure.