Trinomial coefficients, Fibonacci numbers and units of the group [formula omitted

Abstract

Let g be a generator of the cyclic group Cp, p prime. The elements of the form (−1+gtj+g−tj), 0≤j≤p−32−1, which we call 3-supported symmetric, are units of the integral group algebra ZCp. We provide an explicit description of the coefficients in the expansion of positive integral powers of the units −1+g+g−1 as a lacunary sum of trinomial coefficients (Ni and Pan (2018) [9]). Finally, for the particular case p=5 we obtain these coefficients in terms of Fibonacci numbers.