A heptagonal triangular number is simultaneously a heptagonal number 
and a triangular number 
for positive integers 
and 
. Such numbers satisfy
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Substituting 
and 
gives the Pell-like quadratic Diophantine
equation
 |
(3)
|
which has basic solutions 
,
(7, 3), and (18, 8). Additional solutions can be obtained from the unit Pell
equation, and correspond to positive integer
solutions when 
,
(5, 10), (221, 493), (1513, 3382), ... (OEIS A046193
and A039835), corresponding to the heptagonal
triangular numbers 1, 55, 121771, 5720653, 12625478965, ... (OEIS A046194).
