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

TL;DR

This paper introduces an integral representation for Kekulé numbers in benzenoids, evaluates related integrals, and explores conjectures linking double integrals to Smarandache sequences, advancing mathematical understanding of these combinatorial and number sequence concepts.

Contribution

It provides a novel integral representation for Kekulé numbers and evaluates specific integrals, connecting them to Smarandache sequences through conjectures.

Findings

Derived an integral formula for Kekulé numbers in benzenoids.

Evaluated integrals of the form  rac{\u03c7(nx)}{^{2}x +k} dx.

Proposed conjectures linking double integrals to Smarandache sequences.

Abstract

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.