Homology of compact non-orientable surfaces

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology and the topological space/family is compact non-orientable surface
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Statement

Suppose $k$ is a positive integer. We denote by $P_{n}$ (not standard notation, should try to find something) the connected sum of the real projective plane with itself $n$ times, i.e., the connected sum of $n$ copies of the real projective plane.

Unreduced version over the integers

We have:

$H_{k}(P_{n};\mathbb {Z} )=\lbrace {\begin{array}{rl}\mathbb {Z} ,&k=0\\\mathbb {Z} ^{(n-1)/2}\oplus \mathbb {Z} /2\mathbb {Z} ,&k=1,n\ \operatorname {odd} \\\mathbb {Z} ^{(n-2)/2}\oplus \mathbb {Z} /2\mathbb {Z} ,&n\ \operatorname {odd} \\0,&k\geq 2\\\end{array}}$