  In this paper, we compute the $RO(C_n)$-graded coefficient ring of
equivariant cohomology for cyclic groups $C_n$, in the case of Burnside ring
coefficients, and in the case of constant coefficients. We use the invertible
Mackey functors under the box product to reduce the gradings in the computation
from $RO(C_n)$ to those expressable as combinations of $\lambda^d$ for divisors
$d$ of $n$, where $\lambda$ is the inclusion of $C_n$ in $S^1$ as the roots of
unity. We make explicit computations for the geometric fixed points for
Burnside ring coefficients, and in the positive cone for constant coefficients.
The positive cone is also computed for the Burnside ring in the case of prime
power order, and in the case of square free order. Finally, we also make
computations at non-negative gradings for the constant coefficients.
