  Let $\{B_n\}$, $\{B_n(x)\}$ and $\{E_n(x)\}$ be the Bernoulli numbers,
Bernoulli polynomials and Euler polynomials, respectively. In this paper we
mainly establish formulas for $\sum_{6\mid k-3}\binom nkB_{n-k}(x)$,
$\sum_{6\mid k}\binom nkE_{n-k}(x)$ and $\sum_{6\mid k-3}\binom nkm^kB_{n-k}$
in the cases $m=2,3,4$.
