Golombic and Levine sequences

TL;DR

The paper addresses the challenge of computing and understanding far-out terms of Golomb-type (golombic) and Levine sequences by developing a unified algebraic framework. It introduces and analyzes the golombic operator $\Gamma$, the Levine operator $L$, and their generalized forms on words in the free group on the integers, together with Vardi's polynomials $T_n$ and a discrete calculus for functionals. The main contributions include extending the operator theory to ${F\mathbb Z}$, establishing iterative formulas for golombic and Levine numbers, and implementing PARI/GP algorithms that compute large terms (e.g., $\gamma(2)$ up to term $n=19$ and $\ell(2)$ up to term $n=20$). These results illuminate structural connections between Golomb's and Levine's sequences, enable practical large-term computations, and suggest new avenues for asymptotic and structural analysis of these fast-growing sequences.

Abstract

We investigate and generalise Levine sequences like A011784, A061892 and A061894 and develop an algebraic theory for them. We thereby also cover other fast growing sequences like A014644, which we call golombic due to their strong ties with Golomb's sequence A001462.

Figures (2)

• Figure 1: Calculate $\gamma_1w,\dotsc,\gamma_9w$ from $w=(b_1^{m_1}\!\dotsc b_k^{m_k})\in{F\mathbb Z}$.
• Figure 2: Calculate $\ell_1w,\dotsc,\ell_9w$ from $w=(b_1^{m_1}\!\dotsc b_k^{m_k})\in{F\mathbb Z}$ and $\ell_2w$, $\ell_4w$, $\ell_6w$.

Theorems & Definitions (40)

• Lemma 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof