Triangle Counting

Given rods of length 1, 2, ..., $n$, how many distinct triangles $T(n)$ can be made? Lengths for which

$$l_i=l_j+l_k$$

obviously do not give triangles, but all other combinations of three rods do. The answer is

$$T(n)=\cases{ {\textstyle{1\over 24}}n(n-2)(2n-5) & for $n$\... ...r {\textstyle{1\over 24}}(n-1)(n-3)(2n-1) & for $n$\ odd.\cr}$$

The values for $n=1$, 2, ...are 0, 0, 0, 1, 3, 7, 13, 22, 34, 50, ... (Sloane's A002623). Somewhat surprisingly, this sequence is also given by the Generating Function

$$f(x)={x^4\over(1-x)^3(1-x^2)}=x^4+3x^5+7x^6+13x^7+\ldots.$$

References

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 278-282, 1991.

Sloane, N. J. A. Sequence A002623/M2640 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.