%I A038712
%S A038712 1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,63,1,
%T A038712 3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,127,1,
%U A038712 3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,63,1,3
%N A038712 Let k be highest power of 2 dividing n (A006519); a(n) = 2k-1.
%C A038712 n XOR n-1, i.e. nim-sum of a pair of consecutive numbers.
%C A038712 Denominator of quotient=sigma[2n]/sigma[n]. - Labos E. (labos(AT)ana.sote.hu), 
               Nov 04 2003
%C A038712 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2009: 
               (Start)
%C A038712 a(n) = the Tower of Hanoi disc moved at the n-th move, using standard
%C A038712 moves with discs labeled (1, 3, 7, 15,...) starting from top (smallest 
               = 1). (End)
%C A038712 Equals row sums of triangle A168312 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 22 2009]
%H A038712 R. Zumkeller, <a href="b038712.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A038712 Klaus Brockhaus, <a href="a038712.gif">Illustration of A038712 and A080277</
               a>
%H A038712 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A038712 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A038712 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer 
               generating functions. I. Elementary sequences</a>
%H A038712 <a href="Sindx_Ni.html#Nimsums">Index entries for sequences related to 
               Nim-sums</a>
%F A038712 a(n) = A110654(n-1) XOR A008619(n). - Reinhard Zumkeller, Feb 05 2007
%F A038712 a(n) = 2^A001511(n)-1 = 2*A006519(n)-1 = 2^(A007814(n)+1)-1
%F A038712 Multiplicative with a(2^e)=2^(e+1)-1, a(e^p)=1, p>2. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Nov 06 2001
%F A038712 Sum_{n>0} a(n)*x^n/(1+x^n) = Sum_{n>0} x^n/(1-x^n). Inverse Moebius transform 
               of A048298. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 02 2003
%F A038712 G.f.: Sum(k>=0, 2^k*x^2^k/(1-x^2^k)). a(2n+1) = 1, a(2n) = 2a(n)+1. - 
               Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 15 2003
%F A038712 Equals A130093 * [1, 2, 3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 13 2007
%F A038712 Sum(i=1,n, (-1)^A000120(n-i)*a(i))=(-1)^(A000120(n)-1)*n. [From Vladimir 
               Shevelev (shevelev(AT)bgu.ac.il), Mar 17 2009]
%e A038712 a(6) = 3 because 110 XOR 101 = 11 base 2 = 3
%t A038712 Table[Denominator[DivisorSigma[1, 2*n]/DivisorSigma[1, n]], {n, 1, 128}]
%o A038712 (C:) int a(int n) { return n ^ (n-1); } - Russ Cox (rsc(AT)swtch.com), 
               May 15 2007
%Y A038712 A038713 translated from binary, diagonals of A003987 on either side of 
               main diagonal.
%Y A038712 Cf. A062383. Partial sums give A080277.
%Y A038712 Bisection of A089312. Cf. A088837.
%Y A038712 a(n)-1 is exponent of 2 in A089893(n).
%Y A038712 Cf. A130093.
%Y A038712 This is Guy Steele's sequence GS(6, 2) (see A135416).
%Y A038712 Sequence in context: A021991 A112132 A053381 this_sequence A065745 A117677 
               A038870
%Y A038712 Cf. A168312 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2009]
%Y A038712 Adjacent sequences: A038709 A038710 A038711 this_sequence A038713 A038714 
               A038715
%K A038712 easy,nonn,mult,new
%O A038712 1,2
%A A038712 Henry Bottomley (se16(AT)btinternet.com), May 02 2000