A stella octangula number is a figurate number defined for positive integers 
by
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is an octahedral number, 
is a tetrahedral number,
and 
.
The first few are 1, 14, 51, 124, 245, ... (OEIS A007588).
The generating function for the stella octangula
numbers, as a formal power series or for 
,
is
 |
(3)
|
The only positive square stella octangula numbers are 1 and 9653449, which correspond to 
with 
and 
. Since 
and 
are relatively prime,
a square value requires 
and 
for positive integers 
and 
.
The only positive integer solutions of this Diophantine equation are 
and 
(Steiner and Tzanakis 1991, p. 123), yielding
the two stated values.
." J. Number Th. 37, 123-132, 1991.