Triangle counting in this problem asks for the number 
of nondegenerate triangles
that can be formed by choosing three distinct rods from one rod of each length 1,
2, ..., 
, where 
is an integer. Each choice
of three rods is counted once. If the largest length 
and the other two lengths 
and 
satisfy
 |
(1)
|
then the rods do not form a nondegenerate triangle. All other choices satisfy the triangle inequality and do form a triangle. The answer is
 |
(2)
|
The values for 
, 2, ... are 0, 0, 0, 1, 3,
7, 13, 22, 34, 50, ... (OEIS A002623, with
for 
). The ordinary generating
function 
is
 |
(3)
|
for 
.
