  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.
