An octagonal hexagonal number is a number which is simultaneously octagonal and hexagonal. Let 
denote the 
th octagonal number and
the 
th
hexagonal number, for positive
integers 
and 
.
A number which is both octagonal and hexagonal
satisfies the equation 
, or
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
This gives the second-order Diophantine equation
 |
(3)
|
where 
and 
.
The first few positive integer 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 positive
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).
