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

1,3

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

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A229549 A229804 A241096 * A229805 A104075 A339753

Adjacent sequences: A057132 A057133 A057134 * A057136 A057137 A057138

Keyword

base,nonn

Author

Henry Bottomley, Aug 12 2000

Extensions

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

Status

approved