A036044 BCR(n): write in binary, complement, reverse.

1, 0, 2, 0, 6, 2, 4, 0, 14, 6, 10, 2, 12, 4, 8, 0, 30, 14, 22, 6, 26, 10, 18, 2, 28, 12, 20, 4, 24, 8, 16, 0, 62, 30, 46, 14, 54, 22, 38, 6, 58, 26, 42, 10, 50, 18, 34, 2, 60, 28, 44, 12, 52, 20, 36, 4, 56, 24, 40, 8, 48, 16, 32, 0, 126, 62, 94, 30, 110, 46, 78, 14, 118, 54, 86

Offset

0,3

Comments

a(0) could be considered to be 0 if the binary representation of zero were chosen to be the empty string. - Jason Kimberley, Sep 19 2011

From Bernard Schott, Jun 15 2021: (Start)

Except for a(0) = 1, every term is even.

For each q >= 0, there is one and only one odd number h such that a(n) = 2*q iff n = h*2^m-1 for m >= 1 when q = 0, and for m >= 0 when q >= 1 (see A345401 and some examples below).

a(n) = 0 iff n = 2^m-1 for m >= 1 (Mersenne numbers) (A000225).

a(n) = 2 iff n = 3*2^m-1 for m >= 0 (A153893).

a(n) = 4 iff n = 7*2^m-1 for m >= 0 (A086224).

a(n) = 6 iff n = 5*2^m-1 for m >= 0 (A153894).

a(n) = 8 iff n = 15*2^m-1 for m >= 0 (A196305).

a(n) = 10 iff n = 11*2^m-1 for m >= 0 (A086225).

a(n) = 12 iff n = 13*2^m-1 for m >= 0 (A198274).

For k >= 1, a(n) = 2^k iff n = (2^(k+1)-1)*2^m - 1 for m >= 0.

Explanation for a(n) = 2:

For m >= 0, A153893(m) = 3*2^m-1 -> 1011...11 -> 0100...00 -> 10 -> 2 where 1011...11_2 is 10 followed by m 1's. (End)

Links

Indranil Ghosh, Table of n, a(n) for n = 0..10000 (first 1024 terms from T. D. Noe)

Formula

a(2n) = 2*A059894(n), a(2n+1) = a(2n) - 2^floor(log_2(n)+1). - Ralf Stephan, Aug 21 2003

Example

4 -> 100 -> 011 -> 110 -> 6.

Maple

A036044 := proc(n)
    local bcr ;
    if n = 0 then
        return 1;
    end if;
    convert(n, base, 2) ;
    bcr := [seq(1-i, i=%)] ;
    add(op(-k, bcr)*2^(k-1), k=1..nops(bcr)) ;
end proc:
seq(A036044(n), n=0..200) ; # R. J. Mathar, Nov 06 2017

Mathematica

dtn[ L_ ] := Fold[ 2#1+#2&, 0, L ]; f[ n_ ] := dtn[ Reverse[ 1-IntegerDigits[ n, 2 ] ] ]; Table[ f[ n ], {n, 0, 100} ]
Table[FromDigits[Reverse[IntegerDigits[n, 2]/.{1->0, 0->1}], 2], {n, 0, 80}] (* Harvey P. Dale, Mar 08 2015 *)

Prog

(Haskell)
import Data.List (unfoldr)
a036044 0 = 1
a036044 n = foldl (\v d -> 2 * v + d) 0 (unfoldr bc n) where
   bc 0 = Nothing
   bc x = Just (1 - m, x') where (x', m) = divMod x 2
-- Reinhard Zumkeller, Sep 16 2011
(Magma) A036044:=func<n|n eq 0 select 1 else SequenceToInteger(Reverse([1-b:b in IntegerToSequence(n, 2)]), 2)>; // Jason Kimberley, Sep 19 2011
(PARI) a(n)=fromdigits(Vecrev(apply(n->1-n, binary(n))), 2) \\ Charles R Greathouse IV, Apr 22 2015
(Python)
def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)
print([BCR(n) for n in range(75)]) # Michael S. Branicky, Jun 14 2021
(Python)
def A036044(n): return -int((s:=bin(n)[-1:1:-1]), 2)-1+2**len(s) # Chai Wah Wu, Feb 04 2022

Crossrefs

Cf. A030101, A056539, A345401.

Cf. A035928 (fixed points), A195063, A195064, A195065, A195066.

Indices of terms 0, 2, 4, 6, 8, 10, 12, 14, 18, 22, 26, 30: A000225 \ {0}, A153893, A086224, A153894, A196305, A086225, A198274, A052996\{1,3}, A291557, A198276, A171389, A198275.

Sequence in context: A277681 A140876 A243997 * A335790 A078991 A346790

Adjacent sequences: A036041 A036042 A036043 * A036045 A036046 A036047

Keyword

nonn,easy,base,nice,look

Author

N. J. A. Sloane

Status

approved