%TriIntSides.tex:A Maple One-Liner that Proves George Andrews's Theorem That The Number ... %%a Plain TeX file by Shalosh B. Ekhad %begin macros \def\L{{\cal L}} \baselineskip=14pt \parskip=10pt \def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height 0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule height0.15cm width 0.01cm}} \font\eightrm=cmr8 \font\sixrm=cmr6 \font\eighttt=cmtt8 \magnification=\magstephalf \def\P{{\cal P}} \def\Q{{\cal Q}} \def1円{{\overline{1}}} \def2円{{\overline{2}}} \parindent=0pt \overfullrule=0in \def\Tilde{\char126\relax} \def\frac#1#2{{#1 \over #2}} \nopagenumbers %\headline={\rm \ifodd\pageno \RightHead \else \LeftHead \fi} %\def\RightHead{\centerline{ %Title %}} %\def\LeftHead{ \centerline{Doron Zeilberger}} %end macros \bf \centerline { A Maple One-Line Proof of George Andrews's Formula that Says that the Number } \centerline { of Triangles with Integer Sides Whose Perimeter is $n$ Equals $\{ \frac{n^2}{12} \}-\lfloor \frac{n}{4} \rfloor \lfloor \frac{n+2}{4} \rfloor$ } \rm \bigskip \centerline{ {\it Shalosh B. EKHAD }\footnote{$^1$} {\eightrm \raggedright %\break {\eighttt ShaloshBEkhad@gmail.com} \quad . \hfill \break Exclusively published in: {\eighttt http://www.math.rutgers.edu/\~{}zeilberg/pj.html} and {\eighttt http://arxiv.org/} . \hfill \break Feb. 4 2012. } } {\tt evalb(seq(coeff(taylor(q\^{}3/(1-q\^{}2)/(1-q\^{}3)/(1-q\^{}4),q=0,37),q,i),i=0..36) \hfill\break =seq(round(n\^{}2/12)-trunc(n/4)*trunc((n+2)/4),n=0..36)); } {\bf Comments by Doron Zeilberger} {\bf 1}. The succinct formula of the title was discovered and first proved by George Andrews[A1], in a less-than-one-page cute note, improving a four-page note by Jordan et. al. [JWW]. Andrews's note, while short and sweet, is not self-contained, using results from his classic [A2]. {\bf 2}. Here is a clarification of the above Maple one-line proof. Arrange the lengths of the sides of a typical integer-side triangle in non-increasing order, and write them as $[a+b+c+1,b+c+1,c+1]$ for $a,b,c \geq 0$. By [E] I.20, $(b+c+1)+(c+1)>a+b+c+1,ドル so $c \geq a,ドル so $c=a+t$ for $t \geq 0$. So a generic integer-side triangle can be written as $[a+b+(a+t)+1,b+(a+t)+1,a+t+1]=[2a+b+t+1,a+b+t+1,a+t+1],ドル and hence the number of triangles with integer sides whose perimeter equals $n$ is the coefficient of $q^n$ in $$ \sum_{a,b,t \geq 0} q^{(2a+b+t+1)+(a+b+t+1)+(a+t+1)}=\sum_{a,b,t \geq 0} q^{4a+3t+2b+3} =q^3 \left ( \sum_{a \geq 0} q^{4a} \right ) \left ( \sum_{t \geq 0} q^{3t} \right ) \left ( \sum_{b \geq 0} q^{2b} \right )= $$ $$ \frac{q^3}{(1-q^4)(1-q^3)(1-q^2)} \quad . $$ The coefficient of $q^n$ is obviously a {\it quasi-polynomial} of degree 2ドル$ and ``period'' $lcm (2,3,4)=12,ドル but so is $\{ \frac{n^2}{12} \}-\lfloor \frac{n}{4} \rfloor \lfloor \frac{n+2}{4} \rfloor,ドル hence it suffices to check that the first $(2+1) \cdot 12+1=37$ values match. \halmos {\bf 3}. This is yet another example where ``physical'' (as opposed to ``mathematical'') induction, i.e. checking {\bf finitely} (and not that many!) special cases, constitutes a {\it perfectly rigorous proof}! No need for the ``simple'' argument by ``mathematical'' induction alluded to by Andrews in his note. In this case we were in the {\it quasi-polynomial} {\bf ansatz}. See [Z1][Z2], as well as the future classic [KP]. {\bf References} [A1] G. E. Andrews, {\it A note on partitions and triangles with integer sides}, Amer. Math. Monthly {\bf 86} (1979),477-478. [A2] G. E. Andrews, {\it ``The Theory of Partitions''}, Addison-Wesley, 1976. Reprinted by Cambridge University Press, 1984. First paperback edition, 1998. [E] Euclid, {\it ``The Elements''}, Alexandria University Press, ca. 300 BC. [JWW] J.H. Jordan, R. Walch, and R.J. Wisner, {\it Triangles with integer sides}, Amer. Math. Monthly {\bf 86} (1979), 686-689. [KP] M. Kauers and P. Paule, {\it ``The Concrete Tetrahedron''}, Springer, 2011 \hfill\break {\tt http://www.springer.com/mathematics/analysis/book/978-3-7091-0444-6} \quad . [Z1] D. Zeilberger, {\it Enumerative and Algebraic Combinatorics}, in: ``Princeton Companion to Mathematics'' , (Timothy Gowers, ed.), Princeton University Press, pp. 550-561, 2008. \hfill\break {\tt http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimPDF/enu.pdf} [Z2] D. Zeilberger, {\it An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding}, in: C.S. Calude, ed., ``Randomness \& Complexity, from Leibniz to Chaitin'' World Scientific, Singapore, 2007. \hfill\break {\tt http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimPDF/enquiry.pdf} \end

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