A038189 Bit to left of least significant 1-bit in binary expansion of n.

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1

Offset

0,1

Comments

Characteristic function of A091067.

Image, under the coding i -> floor(i/2), of the fixed point, starting with 0, of the morphism 0 -> 01, 1 -> 02, 2 -> 32, 3 -> 31. - Jeffrey Shallit, May 15 2016

Restricted to the positive integers, completely additive modulo 2. - Peter Munn, Jun 20 2022

References

Jean-Paul Allouche and Jeffrey O. Shallit, Automatic sequences, Cambridge, 2003, sect. 5.1.6

Links

Ivan Panchenko, Table of n, a(n) for n = 0..10000

Michael Gilleland, Some Self-Similar Integer Sequences

Index entries for characteristic functions

Index entries for sequences related to binary expansion of n

Index entries for sequences obtained by enumerating foldings

Formula

a(0) = 0, a(2*n) = a(n) for n>0, a(4*n+1) = 0, a(4*n+3) = 1.

G.f.: Sum_{k>=0} t^3/(1-t^4), where t=x^2^k. Parity of A025480. For n >= 1, a(n) = 1/2 * (1 - (-1)^A025480(n-1)). - Ralf Stephan, Jan 04 2004 [index adjusted by Peter Munn, Jun 22 2022]

a(n) = 1 if Kronecker(-n,m)=Kronecker(m,n) for all m, otherwise a(n)=0. - Michael Somos, Sep 22 2005

a(n) = 1 iff A164677(n) < 0. - M. F. Hasler, Aug 06 2015

For n >= 1, a(n) = A065339(n) mod 2. - Peter Munn, Jun 20 2022

From A.H.M. Smeets, Mar 08 2023: (Start)

a(n+1) = 1 - A014577(n) for n >= 0.

a(n+1) = 2 - A014710(n) for n >= 0.

a(n) = (1 - A034947(n))/2 for n > 0. (End)

Example

a(6) = 1 since 6 = 110 and bit before rightmost 1 is a 1.

Maple

A038189 := proc(n)
    option remember;
    if n = 0 then
        0 ;
    elif type(n, 'even') then
        procname(n/2) ;
    elif modp(n, 4) = 1 then
        0 ;
    else
        1 ;
    end if;
end proc:
seq(A038189(n), n=0..100) ; # R. J. Mathar, Mar 30 2018

Mathematica

f[n_] := Block[{id2 = Join[{0}, IntegerDigits[n, 2]]}, While[ id2[[-1]] == 0, id2 = Most@ id2]; id2[[-2]]]; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Apr 14 2009 and fixed Feb 27 2014 *)
f[n_] := f[n] = Switch[Mod[n, 4], 0, f[n/2], 1, 0, 2, f[n/2], 3, 1]; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Apr 14 2009, fixed Feb 27 2014 *)

Prog

(C) int a(int n) { return (n & ((n&-n)<<1)) ? 1 : 0; } /* from Russ Cox */
(PARI) a(n) = if(n<1, 0, ((n/2^valuation(n, 2)-1)/2)%2) /* Michael Somos, Sep 22 2005 */
(PARI) a(n) = if(n<3, 0, prod(m=1, n, kronecker(-n, m)==kronecker(m, n))) /* Michael Somos, Sep 22 2005 */
(PARI) A038189(n)=bittest(n, valuation(n, 2)+1) \\ M. F. Hasler, Aug 06 2015
(PARI) a(n)=my(h=bitand(n, -n)); n=bitand(n, h<<1); n!=0; \\ Joerg Arndt, Apr 09 2021
(Magma)
function a (n)
  if n eq 0 then return 0; // alternatively,  return 1;
  else while IsEven(n) do n := n div 2; end while; end if;
  return n div 2 mod 2; end function;
  nlo := 0; nhi := 32;
  [a(n) : n in [nlo..nhi] ]; //  Fred Lunnon, Mar 27 2018
(Python)
def A038189(n):
    s = bin(n)[2:]
    m = len(s)
    i = s[::-1].find('1')
    return int(s[m-i-2]) if m-i-2 >= 0 else 0 # Chai Wah Wu, Apr 08 2021

Crossrefs

Cf. A038190, A065339.

A014707(n)=a(n+1). A014577(n)=1-a(n+1).

The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - N. J. A. Sloane, Jul 27 2012

Related sequences A301848, A301849, A301850. - Fred Lunnon, Mar 27 2018

Sequence in context: A288508 A262588 A234577 * A072783 A353478 A353555

Adjacent sequences: A038186 A038187 A038188 * A038190 A038191 A038192

Keyword

nonn,easy,base

Author

Fred Lunnon, Dec 11 1999

Extensions

More terms from David W. Wilson

Definition corrected by Russ Cox and Ralf Stephan, Nov 08 2004

Status

approved