Strongly \(\mathbb {Z}\)-graded algebras or principal circle bundles and associated line bundles or invertible bimodules over a class of generalized Weyl algebras \({\mathcal B}(p;q,0)\) (over a ring of polynomials in one variable) are constructed. The Chern-Connes pairing between the cyclic cohomology of \({\mathcal B}(p;q,0)\) and the isomorphism classes of sections of associated line bundles over \({\mathcal B}(p;q,0)\) is computed, thus demonstrating that these bundles, which are labelled by integers, are non-trivial and mutually non-isomorphic. The constructed strongly \(\mathbb {Z}\)-graded algebras are shown to have Hochschild cohomology reminiscent of that of Calabi-Yau algebras. The paper is supplemented by an observation that a grading by an Abelian group in the middle of a short exact sequence is strong if and only if the induced gradings by the outer groups in the sequence are strong.