Homology of quaternionic projective space

Statement
The homology of quaternionic projective space is given as follows:

$$H_p(\mathbb{H} P^n) = \Z \qquad p=0,4,8,\ldots,4n$$

and is zero otherwise.

Related invariants
The Betti numbers of quaternionic projective space are thus $$1$$ for $$4k$$ with $$0 \le k \le n$$ and $$0$$ elsewhere. Thus, the Euler characteristic is $$n+1$$.

Proof
We use the cell decomposition of quaternionic projective space with one cell each in dimensions $$0,4,8,\ldots$$. The cellular chain groups are thus $$\Z$$ in positions $$0,4,\ldots,4n$$ and 0 elsewhere. This forces the cellular homology groups to also be $$\Z$$ exactly in those positions.