Sequences play an important role in many engineering applications. Searching sequences with desired properties has long been an intriguing but also challenging research topic. This article proposes a novel method, called HpGAN, to search desired sequences algorithmically using generative adversarial networks (GANs). HpGAN is based on the idea of zero-sum game to train a generative model, which can generate sequences with characteristics similar to the training sequences. In HpGAN, we design the Hopfield network as an encoder to avoid the limitations of GAN in generating discrete data. Compared with traditional sequence construction by algebraic tools, HpGAN is particularly suitable for complex problems which are intractable by mathematical analysis. We demonstrate the search capabilities of HpGAN in two applications: 1) HpGAN successfully found many different mutually orthogonal complementary sequence sets (MOCSSs) and optimal odd-length binary Z-complementary pairs (OB-ZCPs) which are not part of the training set. In the literature, both MOCSSs and OB-ZCPs have found wide applications in wireless communications and 2) HpGAN found new sequences which achieve a four-times increase of signal-to-interference ratio-benchmarked against the well-known Legendre sequences-of a mismatched filter (MMF) estimator in pulse compression radar systems. These sequences outperform those found by AlphaSeq.