On Newman polynomials without roots on the unit circle

In this note we give a necessary and sufficient condition on the triplet of nonnegative integers $a<b<c$ for which the Newman polynomial $\sum_{j=0}^a x^j + \sum_{j=b}^c x^j$ has a root on the unit circle. From this condition we derive that for each $d \geq 3$ there is a positive integer $n>d$ such that the Newman polynomial $1+x+\dots+x^{d-2}+x^n$ of length $d$ has no roots on the unit circle.