A class of generalized Tribonacci sequences applied to counting problems

Generalized Tribonacci numbers with the third order linear recurrence with constant coefficients T(k)(n)=T(k)(n−1)+T(k)(n−2)+kT(k)(n−3) for n > 2 are investigated for some sets of the initial triples (t0, t1, t2). In particular, generating functions, the Binet formula and the limit of ratio of consecutive terms T(k)(n+1)/T(k)(n) are discussed. These numbers are related to numbers of path graphs colorings with k+2 colors (or, equivalently, to counting of q-ary sequences of length n for q=k+2) satisfying requirements which follow the problem of degeneration in the Ising model with the second neighbor interactions. It is shown that the results obtained can be considered as the base for considerations of cycle graph colorings (cyclic q-ary sequences). These are counting problems, so t0, t1, t2, and k should be natural numbers, but these sequences can be considered for any real numbers. The special cases k=0,1 lead to the Fibonacci and the usual Tribonacci numbers, respectively, so the results can be applied to binary and ternary sequences.