In base 6, these positive numbers and their squares are palindromic.
1, 2, 7, 37, 43, 217, 259, 1297, 1333, 1519, 1555, 7777, 8029, 9079, 46657, 46873, 47989, 48205, 54439, 54655, 279937, 281449, 287749, 326599, 1679617, 1680913, 1687609, 1688905, 1726309, 1727605, 1959559, 1960855, 10077697, 10086769, 10124569, 10357669
1
Written in base 6, the numbers are 1, 2, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001,… whose squares are 1, 4, 121, 10201, 12321, 1002001, 1234321, 100020001, 102030201, 121242121, 123454321, 10000200001,….
T. D. Noe, Plot of 119 terms
T. D. Noe, Table of 119 terms
Eric W. Weisstein, MathWorld: Palindromic Number
(Mma) makePalindrome[n_Integer, b_Integer, del_] := Module[{c = IntegerDigits[n, b], d}, d = If[del, Join[c, Reverse[Most[c]]], Join[c, Reverse[c]]]; FromDigits[d]]; palindromeQ[n_, b_] := Module[{d = IntegerDigits[n, b]}, d == Reverse[d]]; b = 6; t = {}; Do[Do[Do[d = makePalindrome[i, b, j]; e = FromDigits[IntegerDigits[d], b]; If[palindromeQ[e^2, b], AppendTo[t, e]], {i, b^(n - 1), b^n - 1}], {j, {True, False}}], {n, Floor[0.5 + 10*Log[3]/Log[b]]}]
nonn,base
T. D. Noe, Aug 26 2016