Let n be a positive integer. For each 0 <= j <= n-1 we let C_n^j denote the Cayley graph of the cyclic group Zn with respect to the subset {1,j}. Utilizing the Smith Normal Form process, we give an explicit description of the Grothendieck group of each of the Leavitt path algebras LK(C_n^j) for any field K. Our general method significantly streamlines the approach that was used in previous work to establish this description in the specific case j = 2. Along the way, we give necessary and
sufficient conditions on the pairs (j; n) which yield that this group is infinite. We subsequently focus on the case j = 3, where the structure of this group turns out to be related to a Fibonacci-like sequence, called the Narayana's Cows sequence.
