An octagonal square number is a number which is simultaneously an octagonal number and a square number. Let 
denote the 
th octagonal number and
the 
th
square number, with positive
integers 
and 
. The equation 
is equivalent to
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Defining 
and 
gives the Pell equation
 |
(3)
|
The first few positive solutions are 
,
(7, 4), (26, 15), (97, 56), (362, 209), (1351, 780), .... These give the solutions
, (8/3, 4), (9, 15), (98/3, 56), (121, 209),
..., of which the integer solutions are (1, 1), (9, 15),
(121, 209), (1681, 2911), ... (OEIS A046184
and A028230), corresponding to the octagonal
square numbers 1, 225, 43681, 8473921, 1643897025, ... (OEIS A036428).
