A057148 Palindromes only using 0 and 1 (i.e., base-2 palindromes).

0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, 101101, 110011, 111111, 1000001, 1001001, 1010101, 1011101, 1100011, 1101011, 1110111, 1111111, 10000001, 10011001, 10100101, 10111101, 11000011, 11011011, 11100111, 11111111, 100000001

Offset

1,3

Comments

For each term having fewer than 10 digits, the square will also be a palindrome. - Dmitry Kamenetsky, Oct 21 2008

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Mathematica

(* get NextPalindrome from A029965 *)
Select[ NestList[ NextPalindrome, 0, 11110], Max(AT) IntegerDigits(AT)# < 2 &] (* Robert G. Wilson v *)
Select[FromDigits/@Tuples[{0, 1}, 8], IntegerDigits[#]==Reverse[ IntegerDigits[ #]]&] (* Harvey P. Dale, Apr 20 2015 *)

Prog

(Sage)
[int(n.binary()) for n in (0..220) if Word(n.digits(2)).is_palindrome()] # Peter Luschny, Sep 13 2018
(Python)
from itertools import count, islice, product
def agen(): # generator of terms
    yield from [0, 1]
    for d in count(2):
        for rest in product("01", repeat=d//2-1):
            left = "1" + "".join(rest)
            for mid in [[""], ["0", "1"]][d%2]:
                yield int(left + mid + left[::-1])
print(list(islice(agen(), 32))) # Michael S. Branicky, Mar 29 2022
(Python)
def A057148(n):
    if n == 1: return 0
    a = 1<<n.bit_length()-2
    s = bin(a|(n&a-1))[2:]
    return int(s+(s[::-1] if a&n else s[-2::-1])) # Chai Wah Wu, Jun 10 2024

Crossrefs

Cf. A006995 for sequence translated from binary to decimal. A016116 for number of terms of sequence with n+1 binary digits (0 taken to have no digits).

Cf. A002113, A118594, A118595, A118596, A118597, A118598, A118599, A118600.

Sequence in context: A263609 A333415 A264406 * A076289 A350346 A247647

Adjacent sequences: A057145 A057146 A057147 * A057149 A057150 A057151

Keyword

base,nonn

Author

Henry Bottomley, Aug 14 2000

Status

approved