The number of different triangles which have integer side lengths and perimeter 
is
 |  |  |
(1)
|
 |  | ![[(n^2)/(12)]-|_n/4_||_(n+2)/4_|](/images/equations/IntegerTriangle/Inline7.svg) |
(2)
|
 |  | ![{[(n^2)/(48)] for n even; [((n+3)^2)/(48)] for n odd,](/images/equations/IntegerTriangle/Inline10.svg) |
(3)
|
where 
is the partition function giving the number
of ways of writing 
as a sum of exactly 
terms, ![[x]](/images/equations/IntegerTriangle/Inline14.svg)
is the nearest integer function, and 
is the floor
function (Andrews 1979, Jordan et al. 1979, Honsberger 1985). A slightly
complicated closed form is given by
![T(n)=1/(288)[6n^2+18n-9(2n+3)(-1)^n-1
+36sin(1/2pin)-36cos(1/2pin)+64cos(2/3pin)].](/images/equations/IntegerTriangle/NumberedEquation1.svg) |
(4)
|
The values of 
for 
,
2, ... are 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14,
12, 16, ... (OEIS A005044), which is also Alcuin's sequence padded with two initial 0s.
The generating function for 
is given by
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
also satisfies
 |
(8)
|
It is not known if a triangle with integer sides, triangle medians, and area exists (although there are incorrect proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have shown that there are infinitely many triangles with rational sides (Heronian triangles) with two rational triangle medians (Guy 1994).
