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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...
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