A057136 Palindromes whose square root is a palindrome.

0, 1, 4, 9, 121, 484, 10201, 12321, 14641, 40804, 44944, 1002001, 1234321, 4008004, 100020001, 102030201, 104060401, 121242121, 123454321, 125686521, 400080004, 404090404, 10000200001, 10221412201, 12102420121, 12345654321, 40000800004

Offset: 1

Author

Henry Bottomley, Aug 12 2000

Keywords

• base
• nonn

Comments

Always contain an odd number of digits.

Examples

			a(8) = 14641 since 14641 = 121^2 and 121 is also a palindrome
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (n=1..412 from N. J. A. Sloane based on R. J. Mathar's b-file for A057135)

Crossrefs

Cf. A000290, A002113, A002779, A057135 (the square roots).

Programs

Maple

dmax:= 7: # to get all terms with up to dmax digits
Res:= 0,1,2^2,3^2,11^2,22^2:
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))^2 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))^2 fi end proc, Pe))
  fi;
od:
Res; # Robert Israel, Jun 21 2017

Mathematica

Select[Range[0, 10^6], PalindromeQ[#] && PalindromeQ[#^2] &]^2 (* Robert Price, Apr 26 2019 *)

Formula

a(n) = A057135(n)^2