  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.
