Inspired by the Qur'anic parable of Al-Baqarah (2: 261) and the tillering physiology of wheat, this paper presents a structural exploration of odd-only sequences within the Collatzogin Tree—a directed graph constructed from the forward Collatz function. The tree partitions positive integers by their residue modulo 2^k-1, guaranteeing coverage of all integers by construction. From this tree, we extract odd-only predecessor chains and demonstrate that they obey the recurrence aₓ+₁ = 4aₜ + 1, with total stopping times forming arithmetic progressions with common difference 2. The branching distribution follows the Fibonacci sequence (1, 2, 3, 5, 8, 13, 21, 34). We provide a detailed morphological mapping of the tree nodes and derive closed-form formulas for the odd-only sequences. While this framework does not constitute a proof of the Collatz conjecture, it offers a novel combinatorial and structural lens for understanding its underlying arithmetic patterns.
Ogin Sugianto (Wed,) studied this question.
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