Concatenations of powers and their squares
APL (Dyalog Unicode), 21 bytes (SBCS)

Anonymous infix lambda, taking \$r\$ as left argument and \$n\$ as right argument. Requires ⎕IO←0 (zero-based indexing).
{⍵∊(⊢⍎⍤,⍥⍕¨×⍨)⍺*⍳⌊⍟⍵}

Try it online! (Dyalog Extended as polyfill for version 18.0)
{…} &quot;dfn&quot;, ⍺ is \$r\$ and ⍵ is \$n\$:
 ⍟⍵ natural log of \$n\$ (to avoid overflow)
 ⌊ round that down
 ⍳ ɩntegers zero through one less than that
 ⍺* raise \$r\$ to those powers
 (…) apply the following monadic function to that:
  ×⍨ multiply those with themselves (i.e. square them)
  ⊢…¨ for each unmodified argument and its corresponding square:
   ⍥⍕… stringify the argument and its square before
    ⍤, concatenating them, and then
     ⍎ evaluating the result
 ⍵∊ is the original argument a member of that?