How to calculate $\operatorname{taxicab}(3,8,2)$ (sum of 8 cubes in two different ways) Could someone explain me how I can calculate $\operatorname{taxicab}(3,8,2)$?
$\operatorname{taxicab}(3,8,2)$ is the smallest natural number that can be written in $2$ different ways as a sum of $8$ powers $3$. 
For instance:
\begin{align}\operatorname{taxicab}(4,3,2) &= 2673\\
&= 7^4 + 4^4 + 2^4 (or 2401+256+16)
&= 6^4 + 6^4 + 3^4 (or 1296+1296+81)
\end{align}
How can I calculate $\operatorname{taxicab}(3,8,2)$?
This notion is a generalization of the notion of Taxicab number, which is mentioned in the famous story about Hardy and Ramanujan. See also Proof that $1729$ is the smallest taxicab number

$$\eqalign{132=1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 5^3 (or 1+1+1+1+1+1+1+125) =1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3 (or 1+8+8+8+8+8+27+64)}$$
This can be found by a recursive computation.
Let $N(x,m)$ be the number of different ways to write $x$ as the (unordered) sum of $m$ positive cubes. Then $N(x,m) = M(x,m,\lfloor x^{1/3} \rfloor)$ where $M(x,m,n)$ is the number of different ways to write $x = a_1^3  + \ldots + a_m^3$ with $1 \le a_1 \le \ldots \le a_m \le n$.
$$\eqalign{M(x,0,n) &= \cases{1 & if $x=0$\cr 0 & otherwise}\cr 
M(x,m,n) &= \sum_{y=1}^{\min(n, \lfloor x^{1/3} \rfloor)} M(x - y^3, m-1, y)}$$ 
