Writing an integer as a sum of two square in many ways, with consecutive arguments Let $n\in{\mathbb N}$. I call $n=x_1^2+y_1^2=x_2^2+y_2^2=\ldots x_r^2+y_r^2$
(where $(x_1,y_1),(x_2,y_2),\ldots,(x_r,y_r)$ are distinct uples
in ${\mathbb N}^2$) a multi-decomposition of $n$, of length $r$. If the
$x_1,x_2,\ldots,x_r$ form an arithmetic progression of step $1$, I say
that the multi-decomposition is connected.
For example, 
$$
1105=\left\lbrace\begin{array}{lcl}
31^2 &+& 12^2 \\
32^2 &+& 9^2 \\
33^2 &+& 4^2 \\
\end{array}\right.
$$
is a connected multi-decomposition (CMD in abbreviated notation) of length three
for $n=1105$.
With the help of my computer, I found out that for $n\leq 10^6$ there is
no CMD of length four, and the $n$'s having CMD of length three are exactly
the following : $$25, 1105, 12025, 21025, 66625, 252601, 292825, 751825.$$
My questions are :


*

*Are there CMD's with length larger than three ?

*Is there a simple description of all $n$'s having a CMD of length three ?


Although the answer is not quite on the topic.
$$x_1^2+y_1^2=x_2^2+y_2^2=x_3^2+y_3^2=x_4^2+y_4^2$$
For the equation is possible to write General standard formula - moreover it is symmetric.  Interestingly, so monjo to record for any number of summands.
$$x_1=(a^2+b^2)(t^2+k^2)(n^2+q^2)p$$
$$y_1=(a^2+b^2)(t^2+k^2)(n^2+q^2)s$$
$$x_2=(a^2+b^2)(n^2+q^2)(2tkp-(t^2-k^2)s)$$
$$y_2=(a^2+b^2)(n^2+q^2)((t^2-k^2)p+2tks)$$
$$x_3=(a^2+b^2)(t^2+k^2)(2nqp-(n^2-q^2)s)$$
$$y_3=(a^2+b^2)(t^2+k^2)((n^2-q^2)p+2nqs)$$
$$x_4=(t^2+k^2)(n^2+q^2)(2abp-(a^2-b^2)s)$$
$$y_4=(t^2+k^2)(n^2+q^2)((a^2-b^2)p+2abs)$$
