A263608 Palindromes which are base-3 representations of squares.

0, 1, 11, 121, 10201, 11111, 112211, 122221, 1002001, 1120211, 11022011, 100020001, 101212101, 122111221, 1012112101, 1100220011, 10000200001, 10111011101, 110002200011, 111221122111, 1000002000001, 1001221221001, 1012200022101, 1101202021011, 1221221221221, 10101111110101

Offset: 1

Author

N. J. A. Sloane, Oct 22 2015

Keywords

• nonn
• base

Links

• Robert Israel, Table of n, a(n) for n = 1..143
• G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Crossrefs

Cf. A002778, A003166, A029984, A262607, A029985.

Intersection of A001738 and A118594.

Programs

Maple

rev3:= proc(n) local L,i; L:= convert(n,base,3); add(L[-i]*3^(i-1),i=1..nops(L)) end proc:
c3:= proc(n) local L,i; L:= convert(n,base,3); add(L[i]*10^(i-1),i=1..nops(L)) end proc:
R:= 0,1: count:= 2:
for d from 2 while count < 100 do
    if d::odd then
      V:= select(issqr, [seq(seq(a*3^((d+1)/2) + b*3^((d-1)/2)+rev3(a),b=0..2),a=3^((d-3)/2) .. 3^((d-1)/2)-1)])
    else
      V:= select(issqr, [seq(a*3^(d/2) + rev3(a), a=3^(d/2-1) .. 3^(d/2)-1)]);
    fi;
    count:= count+nops(V);
    R:= R, op(map(c3,V));
od:
R; # Robert Israel, May 19 2024

Extensions

Name edited by Robert Israel, May 19 2024