We wish to extend the results above to find the sum of any expression \(a^n + b^n + c^n\) as a sequence of \(n\) for \(3,4,...\) terms.
\[\begin{matrix} 3,2,2/6,5/3,1/5,18/10,0,0& 1+2^n+3^n & A001550\\ 3,3,2/9,25/9,3/25,72/25,0 & 2^n+3^n+4^n & ... \\ 3,14/3,19/21,435/133,315/551,133/29,0 & 2^n + 5^n + 7^n & ...\\ 3,\frac{a+b+c}{3},\frac{2}{3}\frac{a^2+b^2+c^2-ab-ac-bc}{a+b+c},\frac{3}{2}\frac{a^3b+a^3c+b^3a+b^3c+c^3a+c^3b-2a^2b^2-2a^2c^2-2b^2c^2}{a^3+b^3+c^3-3abc}, \frac{1}{2}\frac{(a-b)^2(a-c)^2(b-c)^2(a+b+c)}{2(a^2+b^2+c^2-ab-bc-ca)(a^3b+a^3c+b^3a+b^3c+c^3a+c^3b-2a^2b^2-2a^2c^2-2b^2c^2)},0 \\ 4,5/2,3/6,6/3,2/5,21/10,3/14,16/7,0,0& 1+2^n+3^n+4^n & A001551\\ \end{matrix}\]