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

0,2

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

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)))'.

Mathematica

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

Prog

(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
(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

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.

Sequence in context: A002127 A357764 A061810 * A031159 A058039 A371349

Adjacent sequences: A048698 A048699 A048700 * A048702 A048703 A048704

Keyword

nonn,base

Author

Antti Karttunen, Mar 07 1999

Extensions

Offset corrected by Reinhard Zumkeller, Oct 21 2011

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

Status

approved