An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences

Abstract

We present an integral representation of Kekulé numbers for $P_2(n)$ benzenoids. Related integrals of the form ${\int }_{-\pi }^{\pi }\frac{\cos (nx)}{{\sin }^{2}x+k}dx$ are evaluated. Conjectures relating double integrals of the form ${\int }_0^m{\int }_{-\pi }^{\pi }\frac{\cos (2nx)}{k+{\sin }^{2}x}dxdk$ to Smarandache sequences are presented.