  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 labeled 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.
