A057135 Palindromes whose square is a palindrome; also palindromes whose sum of squares of digits is less than 10.

0, 1, 2, 3, 11, 22, 101, 111, 121, 202, 212, 1001, 1111, 2002, 10001, 10101, 10201, 11011, 11111, 11211, 20002, 20102, 100001, 101101, 110011, 111111, 200002, 1000001, 1001001, 1002001, 1010101, 1011101, 1012101, 1100011, 1101011, 1102011, 1110111, 1111111

Offset: 1

Author

Henry Bottomley, Aug 12 2000

Keywords

• base
• nonn

Examples

			121 is OK since 121^2=14641 is also a palindrome.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (n=1..412 from R. J. Mathar)
• P. De Geest, Subsets of Palindromic Squares

Crossrefs

Cf. A000290, A002113, A002779, A057136, A128921.

Programs

Maple

dmax:= 7: # to get all terms with up to dmax digits
Res:= 0,1,2,3,11,22:
Po:= [[0],[1],[2],[3]]: Pe:= [[0,0],[1,1],[2,2]]:
for d from 1 to dmax do
  if d::odd then
    Po:= select(t -> add(s^2,s=t) < 10, [seq(seq([i,op(t),i], t=Po),i=0..2)]);
    Res:= Res, op(map(proc(p) if p[1] <> 0 then add(p[i]*10^(i-1),i=1..nops(p)) fi end proc, Po))
  else
    Pe:= select(t -> add(s^2,s=t) < 10, [seq(seq([i,op(t),i], t=Pe),i=0..2)]);
    Res:= Res, op(map(proc(p) if p[1] <> 0 then add(p[i]*10^(i-1),i=1..nops(p)) fi end proc, Pe))
  fi;
od:
Res; # Robert Israel, Jun 21 2017

Mathematica

PalQ[n_] := FromDigits[Reverse[IntegerDigits[n]]] == n; t = {}; Do[
If[PalQ[n] && PalQ[n^2], AppendTo[t, n]], {n, 0, 1200000}]; t (* Jayanta Basu, May 10 2013 *)
Select[Range[0,12*10^5],AllTrue[{#,#^2},PalindromeQ]&](* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 20 2018 *)

PARI

is(n) = digits(n)==Vecrev(digits(n)) && digits(n^2)==Vecrev(digits(n^2)) \\ Felix Fröhlich, Jun 21 2017

Formula

a(n) = sqrt(A057136(n))

Extensions

1001001 inserted by R. J. Mathar, Nov 04 2012