A256495 Palindromes i such that 2*i^2 is a palindrome.

0, 1, 2, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 100001, 101101, 110011, 1000001, 1001001, 1010101, 1100011, 10000001, 10011001, 10100101, 11000011, 100000001, 100010001, 100101001, 101000101, 110000011, 1000000001, 1000110001, 1001001001, 1010000101

Offset: 1

Author

Bui Quang Tuan, Mar 31 2015

Keywords

• nonn
• base

Comments

Subsequence of palindromes of A256437.

The sequence contains all positive integers of the form: m*10^(i + NumberOfDigit(m)) + m where i is any nonnegative integer and m is any term of A000533.

Also contains 1 + 10^i and 1 + 10^i + 10^(2*i) for all i >= 1. Are there any members with more than four 1's, or any members other than 2 with digits other than 0's and 1's? - Robert Israel, Apr 13 2015

Examples

			Palindrome 11 is in the sequence because 2*11^2 = 242, a palindrome.
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..91

Crossrefs

Cf. A256437.

Programs

Maple

dmax:= 11: # to get all terms with at most dmax digits
revdigs:= proc(n)
  local L,i;
  L:= convert(n,base,10);
  add(10^(i-1)*L[-i],i=1..nops(L));
end proc:
filter:= proc(n) local L;
  L:= convert(2*n^2,base,10);
  L = ListTools:-Reverse(L)
end proc:
A:= {}:
for d from 1 to dmax do
  if d::even then
     A:= A union select(filter, {seq(10^(d/2)*x + revdigs(x), x=10^(d/2-1)..10^(d/2)-1)})
  else
     m:= (d-1)/2;
     A:= A union select(filter, {seq(seq(10^(m+1)*x + y*10^m + revdigs(x), y=0..9),x=10^(m-1)..10^m-1)})
  fi
od:
A;  # if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 13 2015

Mathematica

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; Select[
Range@ 10000000, palQ@ # && palQ[#^2 + FromDigits[Reverse@ IntegerDigits@ #]^2] &] (* Michael De Vlieger, Mar 31 2015 *)
Select[Range[0,10101*10^5],AllTrue[{#,2#^2},PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 26 2020 *)

PARI

ispal(n) = my(d = digits(n)); Vecrev(d) == d;
lista(nn) = {for (n=0, nn, if (ispal(n) && ispal(2*n^2), print1(n, ", ")););} \\ Michel Marcus, Mar 31 2015

Extensions

a(19)-a(22) from Michel Marcus, Mar 31 2015

a(23)-a(31) from Lars Blomberg, Apr 13 2015