On the lacunary sum of trinomial coefficients

The trinomial coefficient (nk)2 is given by ∑k=−nn(nk)2xk=(1+x+x−1)n.In this paper, we obtain the explicit formulas for the lacunary sum ∑−n≤k≤nk≡r(modm)(nk)2.For example, ∑−n≤k≤nk≡1(mod12)(nk)2=2n+3n−(−1)n+6Hn12,where H0=0,H1=1 and Hn=2Hn−1+2Hn−2 for n ≥ 2.