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

Author

Fred Lunnon, Dec 11 1999

Keywords

• nonn
• easy
• base

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

If a(n) is defined as 1-a(-n) for all n<0, then a(n) = a(2*n), a(4*n+1) = 0, a(4*n+3) = 1 for all n in Z. - Michael Somos, Oct 05 2024

Examples

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

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

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

Programs

C

int a(int n) { return (n & ((n&-n)<<1)) ? 1 : 0; } /* from Russ Cox */

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

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 *)
a[ n_] := If[n==0, 0, Mod[(n/2^IntegerExponent[n, 2]-1)/2, 2]]; (* Michael Somos, Oct 05 2024 *)

PARI

{a(n) = if(n==0, 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

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

Python

def a(n): return 1 if (n & ((n&-n)<<1)) else 0
print([a(n) for n in range(108)]) # Michael S. Branicky, Feb 06 2025 after Russ Cox

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)

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - Amiram Eldar, Aug 30 2024

Extensions

More terms from David W. Wilson

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