cp's OEIS Frontend

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

%I A038189 #112 Feb 06 2025 23:08:27
%S A038189 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,
%T A038189 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,
%U A038189 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
%N A038189 Bit to left of least significant 1-bit in binary expansion of n.
%C A038189 Characteristic function of A091067.
%C A038189 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
%C A038189 Restricted to the positive integers, completely additive modulo 2. - _Peter Munn_, Jun 20 2022
%C A038189 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
%D A038189 Jean-Paul Allouche and Jeffrey O. Shallit, Automatic sequences, Cambridge, 2003, sect. 5.1.6
%H A038189 Ivan Panchenko, <a href="/A038189/b038189.txt">Table of n, a(n) for n = 0..10000</a>
%H A038189 Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A038189 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A038189 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A038189 <a href="/index/Fo#fold">Index entries for sequences obtained by enumerating foldings</a>
%F A038189 a(0) = 0, a(2*n) = a(n) for n>0, a(4*n+1) = 0, a(4*n+3) = 1.
%F A038189 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]
%F A038189 a(n) = 1 if Kronecker(-n,m)=Kronecker(m,n) for all m, otherwise a(n)=0. - _Michael Somos_, Sep 22 2005
%F A038189 a(n) = 1 iff A164677(n) < 0. - _M. F. Hasler_, Aug 06 2015
%F A038189 For n >= 1, a(n) = A065339(n) mod 2. - _Peter Munn_, Jun 20 2022
%F A038189 From _A.H.M. Smeets_, Mar 08 2023: (Start)
%F A038189 a(n+1) = 1 - A014577(n) for n >= 0.
%F A038189 a(n+1) = 2 - A014710(n) for n >= 0.
%F A038189 a(n) = (1 - A034947(n))/2 for n > 0. (End)
%F A038189 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - _Amiram Eldar_, Aug 30 2024
%e A038189 a(6) = 1 since 6 = 110 and bit before rightmost 1 is a 1.
%p A038189 A038189 := proc(n)
%p A038189     option remember;
%p A038189     if n = 0 then
%p A038189         0 ;
%p A038189     elif type(n,'even') then
%p A038189         procname(n/2) ;
%p A038189     elif modp(n,4) = 1 then
%p A038189         0 ;
%p A038189     else
%p A038189         1 ;
%p A038189     end if;
%p A038189 end proc:
%p A038189 seq(A038189(n),n=0..100) ; # _R. J. Mathar_, Mar 30 2018
%t A038189 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 *)
%t A038189 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 *)
%t A038189 a[ n_] := If[n==0, 0, Mod[(n/2^IntegerExponent[n, 2]-1)/2, 2]]; (* _Michael Somos_, Oct 05 2024 *)
%o A038189 (C) int a(int n) { return (n & ((n&-n)<<1)) ? 1 : 0; } /* from _Russ Cox_ */
%o A038189 (PARI) {a(n) = if(n==0, 0, ((n/2^valuation(n,2)-1)/2)%2)}; /* _Michael Somos_, Sep 22 2005 */
%o A038189 (PARI) a(n) = if(n<3, 0, prod(m=1,n, kronecker(-n,m)==kronecker(m,n))) /* _Michael Somos_, Sep 22 2005 */
%o A038189 (PARI) A038189(n)=bittest(n,valuation(n,2)+1) \\ _M. F. Hasler_, Aug 06 2015
%o A038189 (PARI) a(n)=my(h=bitand(n,-n));n=bitand(n,h<<1);n!=0; \\ _Joerg Arndt_, Apr 09 2021
%o A038189 (Magma)
%o A038189 function a (n)
%o A038189   if n eq 0 then return 0; // alternatively,  return 1;
%o A038189   else while IsEven(n) do n := n div 2; end while; end if;
%o A038189   return n div 2 mod 2; end function;
%o A038189   nlo := 0; nhi := 32;
%o A038189   [a(n) : n in [nlo..nhi] ]; //  _Fred Lunnon_, Mar 27 2018
%o A038189 (Python)
%o A038189 def A038189(n):
%o A038189     s = bin(n)[2:]
%o A038189     m = len(s)
%o A038189     i = s[::-1].find('1')
%o A038189     return int(s[m-i-2]) if m-i-2 >= 0 else 0 # _Chai Wah Wu_, Apr 08 2021
%o A038189 (Python)
%o A038189 def a(n): return 1 if (n & ((n&-n)<<1)) else 0
%o A038189 print([a(n) for n in range(108)]) # _Michael S. Branicky_, Feb 06 2025 after _Russ Cox_
%Y A038189 Cf. A038190, A065339.
%Y A038189 A014707(n)=a(n+1). A014577(n)=1-a(n+1).
%Y A038189 The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - _N. J. A. Sloane_, Jul 27 2012
%Y A038189 Related sequences A301848, A301849, A301850. - _Fred Lunnon_, Mar 27 2018
%K A038189 nonn,easy,base
%O A038189 0,1
%A A038189 _Fred Lunnon_, Dec 11 1999
%E A038189 More terms from _David W. Wilson_
%E A038189 Definition corrected by _Russ Cox_ and _Ralf Stephan_, Nov 08 2004