On the entropy of spanning trees on a large triangular lattice

The double integral representing the entropy S_{tri} 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 S_{tri} = [1/(2 Pi)]^2 \int_0^{2 Pi} d\theta \int_0^{2 Pi} 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} - ...)