Integer Triangle

The number of different triangles which have integer side lengths and perimeter is

			(1)
			(2)
			(3)

where is the partition function giving the number of ways of writing as a sum of exactly terms, 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

(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).

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