Archive - Nov 20, 2008

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Small summation problem

Posted November 20th, 2008 by Mgccl
in

\sum _{i=1}^{n-2}\sum _{j=1}^{n-1-i} (n-i-j) = 2\sum _{i=a}^{b}\sum _{j=c}^{d} (n-i-j)
n is even.
Find a,b,c,d, where a,b,c,d are all linear combination of n/2, i and 1.

I stumbled though this problem because I was trying to solve a even harder one
2\sum _{i=a}^{b}\sum _{j=c}^{d} (n-i-j)f(i,j) = \sum _{i=1}^{n-2}\sum _{j=1}^{ n-1-i} (n-i-j)|f(i,j)| where f(i,j) = -f(j,i)

I don't know the difficulty of this question, but I don't have any technique for solving problem like this. So I did exactly what the question asked. Decompose the variables into linear combinations of n/2, i and 1 and brute force all the solutions.
2\sum _{i=k_1\frac{n}{2}+c_1}^{k_2\frac{n}{2}+c_2}\sum _{j=k_3\frac{n}{2}+m_1i+ c_3}^{k_4\frac{n}{2}+m_2i+c_4} (n-i-j) = \sum _{i=1}^{n-2}\sum _{j=1}^{ n-1-i} (n-i-j)

There are a lot of solutions. I put an restriction on all the coefficients. But I found a particular nice one:
2\sum _{i=1}^{\frac{n}{2}}\sum _{j=1}^{\frac{n}{2}-i} (n-i-j) = \sum _{i=1}^{n-2}\sum _{j=1}^{ n-1-i} (n-i-j)

And... I don't like my current MimeTex system. The output equations looks ugly.
I need to find other servers. Currently MathBin is nice and fast, but it is created to be a pastebin. Ahh. I want a delicate server. Or leech from CodeCogs LaTeX Equation editor. Wordpress one is good. but blocked in China.

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