Ansatz in a Nutshell: A comprehensive step-by-step guide to polynomial, $C$-finite, holonomic, and $C^2$-finite sequences

Abstract: Given 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 $C^2$-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 $C^2$-finite sequences. We hope the readers will enjoy the journey through our unifying framework for the study of ansatz.