A number which is simultaneously octagonal and hexagonal. Let 
denote the 
th octagonal number and
the 
th
hexagonal number, then a number which is both
octagonal and hexagonal satisfies the equation 
, or
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Therefore, defining
 |  |  |
(3)
|
 |  |  |
(4)
|
gives the second-order Diophantine equation
 |
(5)
|
The first few solutions are 
, (4, 3), (16, 13), (38, 31), (158, 129), (376, 307),
.... These give the solutions 
, (1, 1), (3, 7/2), (20/3, 8), (80/3, 65/2),
(63, 77), ..., of which the integer solutions are (1, 1), (63, 77), (6141, 7521),
(601723, 736957), ... (OEIS A046190 and A046191), corresponding to the octagonal hexagonal
numbers 1, 11781, 113123361, 1086210502741, ... (OEIS A046192).
