  Let $ p_{i, n}, i=1,..., N(n)=2^{n-1} $ be the lengths of intervals between
the neighboring fractions of Brocot sequence $ F_n $. We obtain an asymptotic
formula for $\sigma_{\beta}(F_n)=\sum_{i=1}^{N(n)} p_{i, n}^{\beta} $,which
improves known estimates.
