A057148 - OEIS

%I #42 Jun 11 2024 01:41:19

%S 0,1,11,101,111,1001,1111,10001,10101,11011,11111,100001,101101,

%T 110011,111111,1000001,1001001,1010101,1011101,1100011,1101011,

%U 1110111,1111111,10000001,10011001,10100101,10111101,11000011,11011011,11100111,11111111,100000001

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

%C For each term having fewer than 10 digits, the square will also be a palindrome. - _Dmitry Kamenetsky_, Oct 21 2008

%H Ray Chandler, <a href="/A057148/b057148.txt">Table of n, a(n) for n = 1..10000</a>

%t (* get NextPalindrome from A029965 *)

%t Select[ NestList[ NextPalindrome, 0, 11110], Max(AT) IntegerDigits(AT)# < 2 &] (* _Robert G. Wilson v_ *)

%t Select[FromDigits/@Tuples[{0,1},8],IntegerDigits[#]==Reverse[ IntegerDigits[ #]]&] (* _Harvey P. Dale_, Apr 20 2015 *)

%o (Sage)

%o [int(n.binary()) for n in (0..220) if Word(n.digits(2)).is_palindrome()] # _Peter Luschny_, Sep 13 2018

%o (Python)

%o from itertools import count, islice, product

%o def agen(): # generator of terms

%o     yield from [0, 1]

%o     for d in count(2):

%o         for rest in product("01", repeat=d//2-1):

%o             left = "1" + "".join(rest)

%o             for mid in [[""], ["0", "1"]][d%2]:

%o                 yield int(left + mid + left[::-1])

%o print(list(islice(agen(), 32))) # _Michael S. Branicky_, Mar 29 2022

%o (Python)

%o def A057148(n):

%o     if n == 1: return 0

%o     a = 1<<n.bit_length()-2

%o     s = bin(a|(n&a-1))[2:]

%o     return int(s+(s[::-1] if a&n else s[-2::-1])) # _Chai Wah Wu_, Jun 10 2024

%Y 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).

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

%K base,nonn

%O 1,3

%A _Henry Bottomley_, Aug 14 2000