Non-Palindromes in Base 2 by Reverse-and-Add Process

Theorem
Let the number $22$ be expressed in binary: $10110_2$.

When the reverse-and-add process is performed on it repeatedly, it never becomes a palindrome.

Proof
It remains to be shown that a binary number of this form does not become a palindrome.

Let $d_n$ denote $n$ repetitions of a binary digit $d$ in a number.

Thus:
 * $10111010000$

can be expressed as:
 * $101_3010_4$

Beware that the subscript, from here on in, does not denote the number base.

It is to be shown that the reverse-and-add process applied to:
 * $101_n010_{n + 1}$

leads after $4$ iterations to:
 * $101_{n + 1}010_{n + 2}$

Thus:

As neither $101_n010_{n + 1}$ nor $101_{n + 1}010_{n + 2}$ are palindromes, nor are any of the intermediate results, the result follows.