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Triangle Counting

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


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

{\textstyle{1\over 24}}n(n-2)(2n-5) & for $n$\...
{\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



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.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

© 1996-9 Eric W. Weisstein