[ January 15, 2015 ]
n and n_reversed are square numbers
whose roots may also be [ pseudo-] reversible
Andres Molina from Tulsa (email)

1089 is a smaller square number example whereby its reverse is also a square number
(n being not palindromic and without ending with zero).

12303690084 is the smallest square number whose reversal is also a square number
but so that their roots are pseudo-reversals amongst themselves (in base 10 at least).

What are the methods to transform nonreversible pairs into genuine reversibles ?
One way is to replace substrings using value n = – 1 like follows :

09 (leading zero included) can be replaced by 1n : which is in fact 10 – 1
099 (leading zero included) can be replaced by 10n : which is in fact 100 – 1
19 can be replaced by 2n : which is in fact 20 – 1
199 can be replaced by 20n : which is in fact 200 – 1
etc.

Let me clarify with next pair of reversible squares 12303690084 & 48009630321
110922 = 111n22
and
219111 = 22n111
As you can see now the roots 111n22 and 22n111 have changed into reversibles.

Another larger example with 102030003060900000804 & 408000009060300030201
10100990202 = 101010n0202
and
20199010101 = 2020n010101
And hey presto the reversibles 101010n0202 and 2020n010101 pop up.

Can you find more such number pairs whose square roots are [pseudo-] reversals as well ?
Or larger sporadic numbers beyond those listed in A061457 and A156316 ?

When n and R(n) are equal they are of course palindromic.
Please link to Palindromic Squares.