An octagonal triangular number is a number which is simultaneously an octagonal number and a triangular number. Let 
denote the 
th
octagonal number and 
the 
th triangular number,
with positive integers 
and 
. The equation 
is equivalent to
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Defining 
and 
gives the second-order Diophantine equation
 |
(3)
|
The first few positive solutions are 
, (4, 3), (16, 13), (38, 31), (158, 129), (376, 307),
.... These give the solutions 
, (1, 1), (3, 6), (20/3, 15), (80/3, 64), (63,
153), ..., of which the integer solutions are (1, 1),
(3, 6), (63, 153), (261, 638), (6141, 15041), (25543, 62566), (601723, 1473913),
... (OEIS A046181 and A046182),
corresponding to the octagonal triangular numbers 1, 21, 11781, 203841, 113123361,
... (OEIS A046183).
