  This paper is concerned with finite sequences of integers that may be written
as sums of squares of two nonzero integers. We first find infinitely many
integers $n$ such that $n, n+h$ and $n+k$ are all sums of two squares where $h$
and $k$ are two arbitrary integers, and as an immediate corollary obtain, in
parametric terms, three consecutive integers that are sums of two squares.
Similarly we obtain $n$ in parametric terms such that all the four integers $n,
n+1, n+2, n+4$ are sums of two squares. We also find infinitely many integers
$n$ such that all the five integers $n, n+1, n+2, n+4, n+5$ are sums of two
squares, and finally, we find infinitely many arithmetic progressions, with
common difference $4$, of five integers all of which are sums of two squares.
