An integer triangle is a triangle with positive integer side lengths. The number 
of incongruent integer triangles with perimeter 
is
 |  |  |
(1)
|
 |  | ![[(n^2)/(12)]-|_n/4_||_(n+2)/4_|](/images/equations/IntegerTriangle/Inline8.svg) |
(2)
|
 |  | ![{[(n^2)/(48)] for n even; [((n+3)^2)/(48)] for n odd,](/images/equations/IntegerTriangle/Inline11.svg) |
(3)
|
where 
is the partition function giving the number
of ways of writing 
as a sum of exactly 
positive integers, ![[x]](/images/equations/IntegerTriangle/Inline15.svg)
is the nearest
integer function, and 
is the floor function
(Andrews 1979, Jordan et al. 1979, Honsberger 1985). For nonnegative
integers 
,
an explicit formula 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)
|
For integers 
, 
also satisfies
 |
(8)
|
It is not known if an integer triangle whose three triangle medians and area are all integers 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 and area (Heronian triangles) with two rational triangle medians (Guy 1994).
