This experimental study presents some interesting conjectured relations between some integer sequences and certain graph parameters of the family of linear Jaco graphs $J_n(x)$ where $n = 1,2,3,\dots$. It appears that $\textit{Golden ratio}$-like floor function terms play an important role in the analysis of the graph structural properties of the family of linear Jaco graphs. The experimental methodology to obtain the conjectures is indeed trivial. However, it is the author's view that the proofs or disproofs of the conjectures may be challenging.