In this paper we investigate linear three-term recurrence formulae Zn=T(n)Zn-1+U(n)Zn-2(n⩾2) with sequences of integers (T(n))n⩾0 and (U(n))n⩾0, which are ultimately periodic modulo m, e.g.(T(n)modm)n⩾0=(a0,a1,a2,…,aρ,T1,T2,…,Tw¯)(U(n)modm)n⩾0=(b0,b1,b2,…,bρ,U1,U2,…,Uw¯).In a former paper of this journal the authors computed explicitly the coefficients of a linear three-term recurrence formula for zn=Zrn+i with 0⩽i<r, when (T(n))n⩾0 and (U(n))n⩾0 belong to regular or non-regular Hurwitz-type continued fraction expansions. Using this result we show now that the sequence (Zn)n⩾0 is ultimately periodic modulo m. As a consequence, for Hurwitz-type continued fraction expansions α=[a0;T1(k),…,Tr(k)¯]k=1∞ or α=[a0;a1,T1(k),…,Tr(k)¯]k=1∞ with polynomials T1≠const.,T2,…,Tr we deduce for all positive integers a and m that liminfqq‖qα‖=0, where q≡amodm and ‖·‖ denotes the distance from the nearest integer. Finally, we are particularly interested in the recurrence formula Zn=(an+b)Zn-1+cZn-2 (n⩾2) and compute the length of a period of the sequence (Znmodm)n⩾0, when (a,m) divides b, and (c,m)=1. This generalizes former results of the authors dealing with regular continued fraction expansions of the numbers exp(1/s) for integers s⩾1.