A Maple One-Line Proof of George Andrews's Formula that Says that the Number of Triangles with Integer Sides Whose Perimeter is n Equals {n2/12} -[n/4][(n+2)/4]

By Shalosh B. EKHAD

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(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org)

First Written: Feb. 6, 2012

Last Update of this webpage (but not of article): Oct. 26, 2012.

Yet another example where "physical" ("incomplete") induction (i.e. only checking finitely many special cases) gives a fully rigorous proof, notwithstanding what your "Intro To Proofs" prof told you!
Added Oct. 26, 2012 (by DZ): Günter Ziegler presented this proof in a students' seminar, also attended by faculty members, and his colleague Günter Rote noticed that Shalosh's proof was longer than necessary! Of course, 36 values suffice (i.e. you only need to check the coeff. of qi from 0 to 35) to prove the identity. This cuts the proof by more than 3 percents!

Added Feb. 9, 2012 (by D. Zeilberger): At least George Andrews only took less than one page, back in 1979, in a modest Monthly Note, but see the recent, Feb. 2012 issue of the Amer. Math. Monthly FULL LENGTH ARTICLE, pages 115-121, that repeated this triviality, and the same Shalosh proof can be applied to its main "theorem" (theorem 1). By a funny coincidence, five minutes after I got this issue, stepped in Edinah Gnang and showed me the Unfair, and intentionally sarcastic report from Monthly Editor Scott Chapman rejecting Edinah Gnang and Chetan Tonde's beautiful submission. They obnoxiously, and self-righteously, state that the Monthly intends to make hard math easy, while the Gnang-Tonde article makes easy math hard! Good intentions, but I am sure that the AMM 119(2), 115-121, article is just one example out of many Monthly article that make trivial things look "deep". On the other hand the rejected submission is full of beautiful insights, missed by the "members" of the "editorial board".

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