From a3b2cbb073d12f7f1d5890e483545d9bf10588d0 Mon Sep 17 00:00:00 2001 From: Roope Salmi Date: Thu, 7 Mar 2019 13:27:49 +0200 Subject: [PATCH] Properly sized brackets in Chapter 7 formula --- chapter07.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/chapter07.tex b/chapter07.tex index 70fd873..120513d 100644 --- a/chapter07.tex +++ b/chapter07.tex @@ -1039,7 +1039,7 @@ As a final note, there is also a surprising direct formula for calculating the number of tilings\footnote{Surprisingly, this formula was discovered in 1961 by two research teams \cite{kas61,tem61} 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 the number of tilings in $O(nm)$ time, but since the answer is a product of real numbers,