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For $a,b\in\mathbb{N}_0$, we consider $(an+b)$-color compositions of apositive integer $\nu$ for which each part of size $n$ admits $an+b$ colors. Westudy these compositions from the enumerative point of view and give a formulafor the number of $(an+b)$-color compositions of $\nu$ with $k$ parts. Ourformula is obtained in two different ways: 1) by means of algebraic propertiesof partial Bell polynomials, and 2) through a bijection to a certain family ofweak compositions that we call domino compositions. We also discuss two caseswhen $b$ is negative and give corresponding combinatorial interpretations.