Krityakierne T.Thanatipanonda T.A.Mahidol University2024-12-032024-12-032024-01-01Springer Proceedings in Mathematics and Statistics Vol.471 (2024) , 255-29721941009https://repository.li.mahidol.ac.th/handle/20.500.14594/102256Given a sequence 1, 1, 5, 23, 135, 925, 7285, 64755, 641075, 6993545, 83339745,…, how can we guess a formula for it? This article will quickly walk you through the concept of ansatz for classes of polynomial, C-finite, holonomic, and the most recent addition C2-finite sequences. For each of these classes, we discuss in detail various aspects of the guess and check, generating functions, closure properties, and closed-form solutions. Every theorem is presented with an accessible proof, followed by several examples intended to motivate the development of the theories. Each example is accompanied by a Maple program with the purpose of demonstrating use of the program in solving problems in this area. While this work aims to give a comprehensive review of existing ansatzes, we also systematically fill a research gap in the literature by providing theoretical and numerical results for the C2-finite sequences.MathematicsConference PaperSCOPUS10.1007/978-3-031-69706-7_112-s2.0-8521032022721941017