Methodological Generalization of the Collatz Sequences to (1 + 2^k )n + S_k(n) with Computational Verification for k = 1 up to k = 51

Several attempts have been made to generalize the Collatz sequence 3n+1, but many of them produced sequences that lack the essential structural properties of the original Collatz dynamics. Among these, the most promising known generalization is the one proposed in 2022 by Naouel Boulkaboul [2], which takes the form 3n + 3^k and leads sequences to converge toward 3^k. In this work, we propose a new methodological generalization of the Collatz conjecture based on the transformation (1 + 2^k)n + S_k(n), where Sk(n) is a function specifically designed to preserve the singularity and the core behavior of the original sequence. We demonstrate that this generalization retains key features such as singularity patterns and trivial cycles. Furthermore, we conduct a computational verification for k = 1 to k = 51, supporting the conjecture that all such generalized sequences converge to 1. This generalization opens the door to new strategies for understanding and potentially proving the Collatz conjecture in its broader form.