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Properly sized brackets in Chapter 7 formula

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Roope Salmi 2019-03-07 13:27:49 +02:00 committed by GitHub
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@ -1039,7 +1039,7 @@ As a final note, there is also a surprising direct formula
for calculating the number of tilings\footnote{Surprisingly, for calculating the number of tilings\footnote{Surprisingly,
this formula was discovered in 1961 by two research teams \cite{kas61,tem61} this formula was discovered in 1961 by two research teams \cite{kas61,tem61}
that worked independently.}: that worked independently.}:
\[ \prod_{a=1}^{\lceil n/2 \rceil} \prod_{b=1}^{\lceil m/2 \rceil} 4 \cdot (\cos^2 \frac{\pi a}{n + 1} + \cos^2 \frac{\pi b}{m+1})\] \[ \prod_{a=1}^{\lceil n/2 \rceil} \prod_{b=1}^{\lceil m/2 \rceil} 4 \cdot \left(\cos^2 \frac{\pi a}{n + 1} + \cos^2 \frac{\pi b}{m+1}\right)\]
This formula is very efficient, because it calculates This formula is very efficient, because it calculates
the number of tilings in $O(nm)$ time, the number of tilings in $O(nm)$ time,
but since the answer is a product of real numbers, but since the answer is a product of real numbers,