A New Type of Inclusion Exclusion Principle for Sequences and Asymptotic Formulas for ζ(k) Original Research Article

Let {a(n)}∞n = 0 be a sequence defined by the recurrence a(n + 2) = Aa(n + 1) + Ba(n), where a(0) = 0 and a(1), A, B are non-zero integers satisfying A2 + 4B ≠ 0. Assume that the ratio two roots of x2 − Ax − B is not a root of unity. It is proved, for each positive integer l, that [formula] where [formula] and κ is a constant, depends only on A and B. This formula is an improvement and a generalization of the results of S. Akiyama (1990, J. Number Theory36, 328-331). In the proof of this formula, we propose a new type of inclusion exclusion principle in a multiplicative sense which itself is of interest.