An Integral Representation of Kekul\'e Numbers, and Double Integrals Related to Smarandache Sequences

We present an integral representation of Kekul'{e} 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} dx dk $ to Smarandache sequences are presented.