A048701 List of binary palindromes of even length (written in base 10).

0, 3, 9, 15, 33, 45, 51, 63, 129, 153, 165, 189, 195, 219, 231, 255, 513, 561, 585, 633, 645, 693, 717, 765, 771, 819, 843, 891, 903, 951, 975, 1023, 2049, 2145, 2193, 2289, 2313, 2409, 2457, 2553, 2565, 2661, 2709, 2805, 2829, 2925, 2973, 3069, 3075, 3171, 3219, 3315

Offset: 0

Author

Antti Karttunen, Mar 07 1999

Keywords

• nonn
• base

Comments

A178225(a(n)) = 1. - Reinhard Zumkeller, Oct 21 2011

a(n) is divisible by 3 and it is always an odd number for n > 0. Therefore a(n) is in A016945 for n > 0. - Altug Alkan, Dec 04 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

Crossrefs

See also A048702 = this sequence divided by 3, A048700 = binary palindromes of odd length, A006995 = all binary palindromes, A048703 = quaternary (base 4) palindromes of even length.

For first differences see A265026, A265027.

Cf. A030308, A007088, A178225.

Programs

Haskell

a048701 n = foldr (\d v -> 2 * v + d) 0 (reverse bs ++ bs) where
   bs = a030308_row (n)
-- Reinhard Zumkeller, Feb 19 2003, Oct 21 2011

Mathematica

Prepend[Select[Range@ 3315, Reverse@ # == # && EvenQ@ Length@ # &@ IntegerDigits[#, 2] &], 0] (* Michael De Vlieger, Dec 04 2015 *)

PARI

a048701(n) = my(f); f = length(binary(n)) - 1; 2^(f+1)*n + sum(i=0, f, bittest(n, i) * 2^(f-i)); \\ Altug Alkan, Dec 03 2015

Python

def A048701(n):
    s = bin(n)[2:]
    return int(s+s[::-1],2) # Chai Wah Wu, Feb 26 2021

Formula

a(n) = (2^(floor_log_2(n)+1))*n + Sum_{i=0..floor_log_2(n)} '(bit_i(n, i)*(2^(floor_log_2(n)-i)))'.

Extensions

Offset corrected by Reinhard Zumkeller, Oct 21 2011

Offset changed back to 0 by Andrey Zabolotskiy, Dec 26 2022