In base 9, these positive numbers and their squares are palindromic.
1, 2, 10, 20, 82, 91, 100, 164, 730, 820, 1460, 6562, 6643, 6724, 7300, 7381, 7462, 13124, 59050, 59860, 65620, 66430, 118100, 531442, 532171, 532900, 538084, 538813, 539542, 590500, 591229, 591958, 597142, 597871, 1062884, 4782970, 4790260, 4842100
1
Written in base 9, these numbers are 1, 2, 11, 22, 101, 111, 121, 202, 1001, 1111, 2002, 10001, 10101, 10201,… which appear to have only the digits 0, 1, and 2. The squares are 1, 4, 121, 484, 10201, 12321, 14641, 40804, 1002001,….
T. D. Noe, Plot of 231 terms
T. D. Noe, Table of 231 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 = 9; 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