  We analyse the numbers of closed paths of length $k\in\mathbb{N}$ on two
important regular lattices: the hexagonal lattice (also called
$\textit{graphene}$ in chemistry) and its dual triangular lattice. These
numbers form a moment sequence of specific random variables connected to the
distance of a position of a planar random flight (in three steps) from the
origin. Here, we refer to such a random variable as a $\textit{random
eigenvalue}$ of the underlying lattice. Explicit formulas for the probability
density and characteristic functions of these random eigenvalues are given for
both the hexagonal and the triangular lattice. Furthermore, it is proven that
both probability distributions can be approximated by a functional of the
random variable uniformly distributed on increasing intervals $[0,b]$ as
$b\to\infty$. This yields a straightforward method to simulate these random
eigenvalues without generating graphene and triangular lattice graphs. To
demonstrate this approximation, we first prove a key integral identity for a
specific series containing the third powers of the modified Bessel functions
$I_n$ of $n$th order, $n\in\mathbb{Z}$. Such series play a crucial role in
various contexts, in particular, in analysis, combinatorics, and theoretical
physics.
