The Domb Ap'ery-limit and a proof of the Ramanujan Machine conjecture Z2

TL;DR

This paper proves a conjecture related to the Ramanujan Machine by establishing specific limits and sums involving Apéry-like sequences, Domb numbers, and the zeta function, using advanced modular form techniques.

Contribution

It provides a rigorous proof of the Ramanujan Machine conjecture for Z2, connecting Apéry-like sequences, Domb numbers, and modular forms.

Findings

Converges to (7/24)ζ(3) the ratio B_n/D_n.

Sum of 64^n/(n^3 D_n D_{n-1}) equals (56/3)ζ(3).

Establishes Z_2 = 12/(7ζ(3)).

Abstract

We prove that the ratio $B_n/D_n$ of the Ap\'ery-like sequence $B_n$ to the Domb numbers $D_n$ converges to $(7/24)\zeta(3)$, and that $\sum_{n=1}^{\infty} 64^n/(n^3 D_n D_{n-1}) = (56/3)\zeta(3)$. As a corollary we establish the value $Z_2 = 12/(7\zeta(3))$ conjectured by the Ramanujan Machine project. The proof uses level-6 eta products, Atkin--Lehner involutions, and Eichler integrals of weight-4 modular forms.