On searching for solutions of the Diophantine equation $x^3 + y^3 +2z^3 = n$

We propose an efficient search algorithm to solve the equation $x^3+y^3+ 2z^3=n$ for a fixed value of $n>0$. By parametrizing $\vert z\vert$, this algorithm obtains $\vert x\vert$ and $\vert y\vert$ (if they exist) by solving a quadratic equation derived from divisors of $2\vert z\vert^3 \pm n$. Thanks to the use of several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for six values of $n$ below 1000 for which no solution had previously been found. We found three new integer solutions for $n=183, 491$ and 931 in the range of $\vert z\vert \le 5 \cdot 10^7$.