= Solution
Let $a$ be the projective-line loop and $b$ the core loop of the <Möbius band>. The boundary of the <Möbius band> goes twice around its core. Its boundary inclusion is a <cofibration>, so replacing the gluing map by a homotopic cellular degree-$n$ map preserves the homotopy type. More precisely, the band is a <mapping cylinder> of the boundary double covering onto the core circle. After attaching its other end and collapsing the connecting tree, the resulting space has a homotopy-equivalent <CW complex> with one vertex, edges $a,b$, and two two-cells with attaching words
$$
a^2,\qquad a^{-n}b^2.
$$
The first is the projective-plane face; the second records equality of the attached boundary loop $a^n$ and the band boundary loop $b^2$. Taking exponent sums gives the <cellular chain complex>
$$
0\longrightarrow\mathbb Z^2\xrightarrow{\;D\;}\mathbb Z^2\xrightarrow{\;0\;}\mathbb Z\longrightarrow0,\qquad
D=\begin{pmatrix}2&-n\\0&2\end{pmatrix}.
$$
Since $\det D=4$, its kernel is zero. The first invariant factor in the <Smith normal form> is the greatest common divisor of its entries, $d=\gcd(2,n)$, and the second is $4/d$. Hence
$$
\boxed{H_0(X;\mathbb Z)=\mathbb Z,\qquad H_1(X;\mathbb Z)=\begin{cases}\mathbb Z/4,&n\text{ odd},\\(\mathbb Z/2)^2,&n\text{ even},\end{cases}\qquad H_j(X;\mathbb Z)=0\ (j\geq2).}
$$
This includes $n=0$ and negative degrees. The <fundamental group after attaching a Möbius band> is $\langle a,b\mid a^2=1,\ b^2=a^n\rangle$, consistent with the computed abelianization. The calculation is an example of <homology after attaching a Möbius band to the real projective plane>.
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