Let L( y)=0 be a linear homogeneous ordinary differential equation with polynomial coefficients. One of the general problems connected with such an equation is to find all points a (ordinary or singular) and all formal power series ∑ n=0 ∞ c n ( x− a) n which satisfy L( y)=0 and whose coefficient c n — considered as a function of n — has some ‘nice’ properties: for example, c n has an explicit representation in terms of n, or the sequence ( c 0, c 1,…) has many zero elements, and so on. It is possible that such properties appear only eventually (i.e., only for large enough n). We consider two particular cases: 1. ( c 0, c 1,…) is an eventually rational sequence, i.e., c n = R( n) for all large enough n, where R( n) is a rational function of n; 2. ( c 0, c 1,…) is an eventually m-sparse sequence, where m⩾2, i.e., there exists an integer N such that (c n≠0)⇒(n≡N ( mod m)) for all large enough n. Note that those two problems were previously solved only ‘for all n’ rather than ‘for n large enough’, although similar problems connected with polynomial and hypergeometric sequences of coefficients have been solved completely.