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Let $\{B_n\}$, $\{B_n(x)\}$ and $\{E_n(x)\}$ be the Bernoulli numbers,Bernoulli polynomials and Euler polynomials, respectively. In this paper wemainly 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 $m=2,3,4$.

Recurrence formulas with gaps for Bernoulli and Euler polynomials