Sum

$\begin{aligned} S & = \sum_{k=0}^\infty \frac 1{(3k+1)(3k+2)(3k+3)} & \small \color{#3D99F6} \text{By partial fraction decomposition} \\ & = \frac 12 \sum_{k=0}^\infty \left({\color{#3D99F6}\frac 1{3k+1}} - \frac 2{3k+2} + \frac 1{3k+3}\right) \\ & = \frac 12 \left({\color{#3D99F6}\frac 11} - \frac 22 + \frac 13 + {\color{#3D99F6}\frac 14} - \frac 25 + \frac 16 + {\color{#3D99F6}\frac 17} - \frac 28 + \frac 19 + \cdots \right) \\ & = - \left(\frac {\cos \frac {4\pi}3}1 + \frac {\cos 2\pi}2 + \frac {\cos \frac {8\pi}3}3 + \frac {\cos \frac {10\pi}3}4 + \cdots \right) \\ & = - {\color{#D61F06}\Re} \sum_{k=1}^\infty \frac {{\color{#3D99F6}\omega}^{k+1}}k & \small \color{#3D99F6} \text{where }\omega = e^{\frac {2\pi}3i} \text{ the 3rd root of unity.} \\ & = - {\color{#D61F06}\Re} \left( \omega \sum_{k=1}^\infty \frac {\omega^k}k \right) & \small \color{#D61F06} \Re(\cdot) \text{ denotes the real part of a complex number.} \\ & = \Re \left( \omega \ln (1-{\color{#3D99F6}\omega}) \right) & \small \color{#3D99F6} \text{By Euler's formula: }e^{\theta i} = \cos \theta + i\sin \theta \\ & = \Re \left( \omega \ln \left(1-{\color{#3D99F6}\left(-\frac 12+\frac {\sqrt 3}2i \right)}\right) \right) \\ & = \Re \left( \omega \ln \left(\frac 32-\frac {\sqrt 3}2i \right) \right) \\ & = \Re \left( \omega \ln \left(\sqrt 3 e^{-\frac \pi 6i} \right) \right) \\ & = \Re \left(\left(-\frac 12 + \frac {\sqrt 3}2i\right) \left(\frac {\ln 3}2 - \frac \pi 6i \right) \right) \\ & = \boxed{\dfrac {\pi \sqrt 3-3\ln 3}{12}} \end{aligned}$