I am trying to understand twisted coefficients better, because although I technically know the definitions both using modules and group bundles, my intution of the subject doesn't really exist yet. To this end I wanted to understand the groups $H_i(S^1, \mathbb{Z}[t])$ where a generator of $\pi_1(S^1)$ acts on $\mathbb{Z}[t]$ by multiplication by $t$.
Now I know I can just perform the computation because we can find a very agreeable cell structure on the universal cover of $S^1$ and then work on the complex $C_*(\mathbb{R})\otimes_{\mathbb{Z}[\pi_1]}\mathbb{Z}[t]$. But this doesn't satisfy me from an intuitive point of view.
So my question is, is it possible to describe a $\mathbb{Z}$-bundle $E\to S^1$ such that $H_i(S^1, E)\cong H_i(S^1, \mathbb{Z}[t])$? I tried finding some product or disjoint union of the universal cover, but didn't find anything.
Thankss