Sequence of Palindromic Cubes

Sequence
The sequence of positive integers whose cube is palindromic begins:
 * $1, 2, 7, 11, 101, 111, 1001, 2201, 10 \, 001, 10 \, 101, \ldots$

Note that $2201$ is the smallest (and only one one known) which is itself non-palindromic.

The corresponding sequence of palindromic cubes begins:
 * $1, 8, 343, 1331, 1 \, 030 \, 301, 1 \, 367 \, 631, 1 \, 003 \, 003 \, 001, 10 \, 662 \, 526 \, 601, 1 \, 000 \, 300 \, 030 \, 001, \ldots$

Also see

 * Palindromic Cube with Non-Palindromic Root