GCD-Augmented Fibonacci Sequences: Closed Forms for Generalized Seeds

We study the integer sequence defined by the recurrence Cp(n) = Cp(n−1) + Cp(n−2) + gcd(Cp(n−1), Cp(n−2)) with seeds Cp(1) = 1, Cp(2) = p, generalizing OEIS sequence A083658 (the case p = 1). We establish closed-form expressions for two families of seeds. For even p, the sequence is purely geometric with base 2: Cp(n) = (p+2) · 2n−3 for n ≥ 3 (Theorem 1). For odd p with gcd(p, 3) = 1, the odd-indexed and even-indexed subsequences are each geometric with base 3, yielding Cp(2k+1) = (p+2) · 3k−1 and Cp(2k) = 5(p+2) · 3k−3, with the consecutive ratio alternating between 5/3 and 9/5 (Theorems 2–3). We prove a scaling property Cka,kb(n) = k · Ca,b(n) reducing the study of arbitrary seed pairs to coprime seeds (Theorem 4), and characterize an anomalous regime for odd seeds divisible by 3 where the geometric structure breaks down (Proposition 1). All results are verified computationally for seeds p = 1 through 49.