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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 doubleintegrals of the form $\int_{0}^{m} \int_{-\pi}^{\pi} \frac{\cos(2nx)}{k+\sin^{2}x} dx dk $ to Smarandache sequences are presented.

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