  We study the number $P(n)$ of partitions of an integer $n$ into sums of
distinct squares and derive an integral representation of the function $P(n)$.
Using semi-classical and quantum statistical methods, we determine its
asymptotic average part $P_{as}(n)$, deriving higher-order contributions to the
known leading-order expression [M. Tran {\it et al.}, Ann.\ Phys.\ (N.Y.) {\bf
311}, 204 (2004)], which yield a faster convergence to the average values of
the exact $P(n)$. From the Fourier spectrum of $P(n)$ we obtain hints that
integer-valued frequencies belonging to the smallest Pythagorean triples
$(m,p,q)$ of integers with $m^2+p^2=q^2$ play an important role in the
oscillations of $P(n)$. Finally we analyze the oscillating part $\delta
P(n)=P(n)-P_{as}(n)$ in the spirit of semi-classical periodic orbit theory [M.
Brack and R. K. Bhaduri: {\it Semiclassical Physics} (Bolder, Westview Press,
2003)]. A semi-classical trace formula is derived which accurately reproduces
the exact $\delta P(n)$ for $n > \sim 500$ using 10 pairs of `orbits'. For $n >
\sim 4000$ only two pairs of orbits with the frequencies 4 and 5 -- belonging
to the lowest Pythagorean triple (3,4,5) -- are relevant and create the
prominent beating pattern in the oscillations. For $n > \sim 100,000$ the beat
fades away and the oscillations are given by just one pair of orbits with
frequency 4.
