On The Combinatorial Identities And The Mean Values Of Some Interesting Sequences

There are three chapters in this thesis. In the first chapter, we studied the combinatorial properties of triangle sequence, quadrangle sequence and pentagon sequence, and gave three combinatorial identities for them.In the second chapter, we studied the mean value of divisor product function. We using the analytic method studied the asymptotic properties of the mean value Ω(Pd(n)), where Pd(n) be the divisor products of n, and got a sharp asymptotic formula for it.In the third chapter, we studied the mean value of some number functions for square residues, and using the analytic method obtained two interesting asymptotic formulas for ∑d(a(n)) and ∑a(n), where a(n) be the square residues of n.