Summary: We study integer sequences associated with the cyclic graph $C_{r}$ and the complete graph $K_{r}$. Fourier techniques are used to characterize the sequences that count walks of length $n$ on both these families of graphs. In the case of the cyclic graph, we show that these sequences are associated with an induced colouring of Pascal's triangle. This extends previous results concerning the Jacobsthal numbers.

11B83, 11Y55, 05T50, 65T50

integer sequences, jacobsthal numbers, Pascal's triangle, cyclic graphs, circulant matrices, discrete Fourier transform