On the Entropy of Spanning Trees on a Large Triangular Lattice

The double integral representing the entropy Stri of spanning trees on a large triangular lattice is evaluated using two different methods, one algebraic and one graphical. Both methods lead to the same result \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ S_{\rm tri} = (4 \pi^2)^{-1} \int_{0}^{2 \pi} d \theta \int^{2 \pi}_{0}d \phi \ln {[}6-2 \cos \theta-2 \cos \phi-2 \cos (\theta + \phi){]} = (3 \sqrt{3}/\pi)(1-5^{-2} + 7^{-2} - 11^{-2} + 13^{-2}-\ldots).$$\end{document}