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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 walkyou through the concept of ansatz for classes of polynomial, $C$-finite,holonomic, and the most recent addition $C^2$-finite sequences. For each ofthese classes, we discuss in detail various aspects of the guess and check,generating functions, closure properties, and closed-form solutions. Everytheorem is presented with an accessible proof, followed by several examplesintended to motivate the development of the theories. Each example isaccompanied by a Maple program with the purpose of demonstrating use of theprogram in solving problems in this area. While this work aims to give acomprehensive review of existing ansatzes, we also systematically fill aresearch gap in the literature by providing theoretical and numerical resultsfor the $C^2$-finite sequences. We hope the readers will enjoy the journeythrough our unifying framework for the study of ansatz.

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