${\left(1+\mathrm{x}\right)}^{\mathrm{n}}={\mathrm{p}}_0+{\mathrm{p}}_1\mathrm{x}+{\mathrm{p}}_2{\mathrm{x}}^2+\dots +{\mathrm{p}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}}, \mathrm{then} {\mathrm{p}}_0+{\mathrm{p}}_3+{\mathrm{p}}_6+\dots \dots =$

${(1+\mathrm{x})}^{\mathrm{n}}={\mathrm{p}}_0+{\mathrm{p}}_1\mathrm{x}+{\mathrm{p}}_2{\mathrm{x}}^2+....+{\mathrm{p}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}}$

Sub  $\mathrm{x}=1, \mathrm{w}, {\mathrm{w}}^2$

$\begin{array}{rcl}{2}^{\mathrm{n}}& =& {\mathrm{p}}_0+{\mathrm{p}}_1+{\mathrm{p}}_2+....+{\mathrm{p}}_{\mathrm{n}}\end{array}$

$\begin{array}{rcl}{(1+\mathrm{w})}^{\mathrm{n}}& =& p_0+{\mathrm{p}}_1\mathrm{w}+{\mathrm{p}}_2{\mathrm{w}}^2+....\end{array}$

$\begin{array}{rcl}{(1+{\mathrm{w}}^2)}^{\mathrm{n}}& =& {\mathrm{p}}_0+{\mathrm{p}}_1{\mathrm{w}}^2+{\mathrm{p}}_2\mathrm{w}+....\end{array}$

$\begin{array}{rcl}\therefore 2^{\mathrm{n}}+{(1+\mathrm{w})}^{\mathrm{n}}+{(1+{\mathrm{w}}^2)}^{\mathrm{n}}& =& 3 ({\mathrm{p}}_0+{\mathrm{p}}_3+{\mathrm{p}}_6+....)\end{array}$

$\begin{array}{rcl}{2}^{\mathrm{n}}+{\left(1-\frac{1}{2}+\frac{\mathrm{i}\sqrt{3}}{2}\right)}^{\mathrm{n}}+{\left(1-\frac{1}{2}-\frac{\mathrm{i}\sqrt{3}}{2}\right)}^{\mathrm{n}}& =& 3({\mathrm{p}}_0+{\mathrm{p}}_3+{\mathrm{p}}_6+....)\\ & & \end{array}$

$\begin{array}{rcl}{\mathrm{p}}_0+{\mathrm{p}}_3+{\mathrm{p}}_6+...& =& \frac{1}{3}\left[2^{\mathrm{n}}+{\left(\frac{1}{2}+\frac{\mathrm{i}\sqrt{3}}{2}\right)}^{\mathrm{n}}+\left(\frac{1}{2}-\frac{\mathrm{i}\sqrt{3}}{2}\right)\right]\\ & =& \frac{1}{3}\left(2^{\mathrm{n}}+{\left(\cos \frac{\mathrm{\pi }}{3}+\mathrm{i} \sin \frac{\mathrm{\pi }}{3}\right)}^{\mathrm{n}}+{\left(\cos \frac{\mathrm{\pi }}{3}-\mathrm{i} \sin \frac{\mathrm{\pi }}{3}\right)}^{\mathrm{n}}\right)\\ & =& \frac{1}{3}\left(2^{\mathrm{n}}+\cos \mathrm{n}\frac{\mathrm{\pi }}{3}+\mathrm{i} \sin \mathrm{n}\frac{\mathrm{\pi }}{3}+\cos \mathrm{n}\frac{\mathrm{\pi }}{3}-\mathrm{i} \sin \mathrm{n}\frac{\mathrm{\pi }}{3}\right)\\ & =& \frac{1}{3}\left(2^{\mathrm{n}}+2\cos \mathrm{n}\frac{\mathrm{\pi }}{3}\right)\end{array}$

${\mathrm{p}}_0+{\mathrm{p}}_3+{\mathrm{p}}_6+\dots \dots ==\frac{2}{3}\left[2^{\mathrm{n}-1}+\cos \frac{\mathrm{n\pi }}{3}\right]$