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In this paper, we consider problems related to the network reliability problem restricted to circulant graphs (networks). Let 1≤s1
2<... k≤[n/2] be given integers. An undirected circulant graph, Cns1,s2,...,sk, has n vertices 0, 1, 2, ..., n-1, and for each si (1≤i≤k) and j (0≤j≤n-1) there is an edge between j and j+si mod n. Let T(Cns1,s2,...,sk) stand for the number of spanning trees of Cns1,s2,...,sk. For this special class of graphs, a general and most recent result is obtained by Y. P. Zhang et al (Discrete Mathematics vol. 223, pp.337-350, 2000) where it is shown that T(Cns1,s2,...,sk)=nan2 where an satisfies a linear recurrence relation of order 2sk-1. In this paper we obtain further properties of the numbers an by considering their combinatorial structures. Using these properties we investigate the open problem posed in the Conclusion of Y. P. Zhang et al. We describe our technique and asymptotic properties of the numbers, using examples.