One month ago, Franklin T. Adams-Watters asked about numbers that occur more than twice in A061205. The result was A203924, to which, yesterday, I added a link to my augmented table of 21313 terms. The majority of these are the third and fourth terms of A061205 quadruples: 101556 being the smallest, appearing at positions 156, 273, 372, and 651. There are 10554 quadruples in my less-than-ten-million position search-range, so these account for 21108 of the entries. The remaining 205 are taken from 67 A061205 triples, 24 sextuples, and 7 octuples.
A solution may be said to be "trivial" if all of its position numbers end in zero. (A non-trivial solution allows for an infinite number of trivial ones by multiplying all of its position numbers by the same power of ten.) After looking at the 40 non-trivial A061205 triples in my table, I conjectured that an odd-tuple exists if the product of a number multiplied by its reversal is either the square of a palindrome or, less frequently, the square of ten times a palindrome.
This insight allowed me to create a table of A061205 triples well beyond my original search range and held out the hope of finding a quintuple, if ever the palindrome could be got at in more than one way. Alas, my program is still considerably brute-force-ish: It searches for a palindrome in the square root of a number multiplied by its reversal. It might well be more efficient doing this the other way 'round.