Integers (a,b,c) whose squares form an arithmetic progression — Part 1

Let’s look at ordered triples of integers   $(a, b, c)$   whose squares form an arithmetic progression.

In other words,
$b^2 \; - \; a^2 \; = \; c^2 \; - \; b^2 \; = \; D^2$,     or equivalently
$2 \, b^2 \; = \; a^2 \; + \; c^2$

The solutions are of the form

$a \; = \; \pm k \, (m^2 \; - \; n^2 \; - \; 2 \, m \, n)$
$b \; = \; \pm k \, (m^2 \; + \; n^2)$
$c \; = \; \pm k \, (m^2 \; - \; n^2 \; + \; 2 \, m \, n)$

for any integers   $m$   and   $n$

The ordered triple   $(a, b, c)$   is a primitive arithmetic progression triple if  $k = 1$

$D \; = \; L \, s^2$,   for   $L < 10$

and where   $(m, n) = 1$

although, we still are getting multiples of primitive solutions, as shown in the table below: