An octagonal heptagonal number is a number which is simultaneously octagonal and heptagonal. Let 
denote the 
th octagonal number and
the 
th
heptagonal number, for positive
integers 
and 
. A number which is both octagonal
and heptagonal 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 
,
(7, 4), (73, 40), (157, 86), .... Selecting the pairs that give positive
integer indices yields 
,
(345, 315), (166145, 151669), ... (OEIS A048904
and A048905), corresponding to the octagonal
heptagonal numbers 1, 297045, 69010153345, ... (OEIS A048906).
