The number of different triangles which have integer side lengths and perimeter n is T(n) | = | P(n, 3) - sum_(1<=j<=⌊n/2⌋) P(j, 2) | = | [n^2/12] - ⌊n/4 ⌋⌊(n + 2)/4 ⌋ | = | {[n^2/48] | for n even [(n + 3)^2/48] | for n odd, auto right match where P(n, k) is the partition function giving the number of ways of writing n as a sum of exactly k terms, [x] is the nearest integer function, and ⌊x⌋ is the floor function (Andrews 1979, Jordan et al. 1979, Honsberger 1985).
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