  Let $n$ be a positive integer. For each $0\leq j \leq n-1$ we let $C_n^{j}$
denote Cayley graph for the cyclic group ${\mathbb Z}_n $ with respect to the
subset $\{1, j\}$. For any such pair $(n,j)$ we compute the size of the
Grothendieck group of the Leavitt path algebra $L_K(C_n^j)$; the analysis is
related to a collection of integer sequences described by Haselgrove in the
1940's. When $j=0,1,$ or 2, we are able to extract enough additional
information about the structure of these Grothendieck groups so that we may
apply a Kirchberg-Phillips-type result to explicitly realize the algebras
$L_K(C_n^j)$ as the Leavitt path algebras of graphs having at most three
vertices. The analysis in the $j=2$ case leads us to some perhaps surprising
and apparently nontrivial connections to the classical Fibonacci sequence.
