Program synthesis is the process of generating a computer program following a set of specifications, such as a set of input-output examples. It can be modeled as a search problem in which the search space is the set of all valid programs. As the search space is vast, brute force is usually not feasible, and search heuristics, such as genetic programming, also have difficulty navigating it without guidance. This text presents 2 novel GP algorithms that synthesize pure, typed, and functional programs: HOTGP and Origami. HOTGP uses strong types and a functional grammar, synthesizing Haskell code, with support for higher-order functions, $λ$-functions, and parametric polymorphism. Experimental results show that HOTGP is competitive with the state of the art. Additionally, Origami is an algorithm that tackles the challenge of effectively handling loops and recursion by exploring Recursion Schemes, in which the programs are composed of well-defined templates with only a few parts that need to be synthesized. The first implementation of Origami can synthesize solutions in several Recursion Schemes and data structures, being competitive with other GP methods in the literature, as well as LLMs. The latest version of Origami employs a novel procedure, called AC/DC, designed to improve the search-space exploration. It achieves considerable improvement over its previous version by raising success rates on every problem. Compared to similar methods in the literature, it has the highest count of problems solved with success rates of $100\%$, $\geq 75\%$, and $\geq 25\%$ across all benchmarks. In $18\%$ of all benchmark problems, it stands as the only method to reach $100\%$ success rate, being the first known approach to achieve it on any problem in PSB2. It also demonstrates competitive performance to LLMs, achieving the highest overall win-rate against Copilot among all GP methods.