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$\begingroup$

Amazingly,

$\prod_{j=1}^{\lceil \frac{m}{2} \rceil}\prod_{k=1}^{\lceil \frac{n}{2} \rceil} [4\cos^2(\frac{\pi j}{m+1})+4\cos^2(\frac{\pi k}{n+1})]$

is an integer for any positive integers $m,n$. In fact it is the number of ways to tile an $m$ by $n$ chessboard by dominoes.

I do not ask for a proof of this last fact; I would just like to know if there is an easy way to see that the above product is an integer?

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    I'm afraid that your assertion is false. For n = 2, m = 4 I find that the product is equal to 5(3+sqrt(5))/2. – 2017-01-02
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    When $n=1$ and $m=4$, Mathematica yields $(19+7\sqrt{5})/2$. – 2017-01-02
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    It seems that the product is integer if you let the indices run to n and m, respectively. And this is still not trivial. – 2017-01-02
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    [These notes](http://www.math.cmu.edu/~bwsulliv/domino-tilings.pdf) (pdf warning) give a slightly different expression, with the upper bounds on the products switched. – 2017-01-02
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    Edited switched the indices my mistake. – 2017-01-02
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    I'm not sure, but maybe it'll help to make $cos^2 = \frac{1+cos2}{2}$ – 2017-01-02

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