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%I #103 Mar 08 2023 14:59:52
%S 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 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 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 Bit to left of least significant 1-bit in binary expansion of n.
%C Characteristic function of A091067.
%C 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 Restricted to the positive integers, completely additive modulo 2. - _Peter Munn_, Jun 20 2022
%D Jean-Paul Allouche and Jeffrey O. Shallit, Automatic sequences, Cambridge, 2003, sect. 5.1.6
%H Ivan Panchenko, <a href="/A038189/b038189.txt">Table of n, a(n) for n = 0..10000</a>
%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Fo#fold">Index entries for sequences obtained by enumerating foldings</a>
%F a(0) = 0, a(2*n) = a(n) for n>0, a(4*n+1) = 0, a(4*n+3) = 1.
%F 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 a(n) = 1 if Kronecker(-n,m)=Kronecker(m,n) for all m, otherwise a(n)=0. - _Michael Somos_, Sep 22 2005
%F a(n) = 1 iff A164677(n) < 0. - _M. F. Hasler_, Aug 06 2015
%F For n >= 1, a(n) = A065339(n) mod 2. - _Peter Munn_, Jun 20 2022
%F From _A.H.M. Smeets_, Mar 08 2023: (Start)
%F a(n+1) = 1 - A014577(n) for n >= 0.
%F a(n+1) = 2 - A014710(n) for n >= 0.
%F a(n) = (1 - A034947(n))/2 for n > 0. (End)
%e a(6) = 1 since 6 = 110 and bit before rightmost 1 is a 1.
%p A038189 := proc(n)
%p option remember;
%p if n = 0 then
%p 0 ;
%p elif type(n,'even') then
%p procname(n/2) ;
%p elif modp(n,4) = 1 then
%p 0 ;
%p else
%p 1 ;
%p end if;
%p end proc:
%p seq(A038189(n),n=0..100) ; # _R. J. Mathar_, Mar 30 2018
%t 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 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 *)
%o (C) int a(int n) { return (n & ((n&-n)<<1)) ? 1 : 0; } /* from _Russ Cox_ */
%o (PARI) a(n) = if(n<1, 0, ((n/2^valuation(n,2)-1)/2)%2) /* _Michael Somos_, Sep 22 2005 */
%o (PARI) a(n) = if(n<3, 0, prod(m=1,n, kronecker(-n,m)==kronecker(m,n))) /* _Michael Somos_, Sep 22 2005 */
%o (PARI) A038189(n)=bittest(n,valuation(n,2)+1) \\ _M. F. Hasler_, Aug 06 2015
%o (PARI) a(n)=my(h=bitand(n,-n));n=bitand(n,h<<1);n!=0; \\ _Joerg Arndt_, Apr 09 2021
%o (Magma)
%o function a (n)
%o if n eq 0 then return 0; // alternatively, return 1;
%o else while IsEven(n) do n := n div 2; end while; end if;
%o return n div 2 mod 2; end function;
%o nlo := 0; nhi := 32;
%o [a(n) : n in [nlo..nhi] ]; // _Fred Lunnon_, Mar 27 2018
%o (Python)
%o def A038189(n):
%o s = bin(n)[2:]
%o m = len(s)
%o i = s[::-1].find('1')
%o return int(s[m-i-2]) if m-i-2 >= 0 else 0 # _Chai Wah Wu_, Apr 08 2021
%Y Cf. A038190, A065339.
%Y A014707(n)=a(n+1). A014577(n)=1-a(n+1).
%Y The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - _N. J. A. Sloane_, Jul 27 2012
%Y Related sequences A301848, A301849, A301850. - _Fred Lunnon_, Mar 27 2018
%K nonn,easy,base
%O 0,1
%A _Fred Lunnon_, Dec 11 1999
%E More terms from _David W. Wilson_
%E Definition corrected by _Russ Cox_ and _Ralf Stephan_, Nov 08 2004
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