%I A038554
%S A038554 0,0,1,0,2,3,1,0,4,5,7,6,2,3,1,0,8,9,11,10,14,15,13,12,4,5,7,6,2,3,1,0,
%T A038554 16,17,19,18,22,23,21,20,28,29,31,30,26,27,25,24,8,9,11,10,14,15,13,12,
%U A038554 4,5,7,6,2,3,1,0,32,33,35,34,38,39,37,36,44,45,47,46,42,43,41,40,56,57
%N A038554 Derivative of n: write n in binary, replace each pair of adjacent bits 
               by their mod 2 sum (a(0)=a(1)=0 by convention). Also n XOR (n shift 
               1).
%C A038554 Comment from Antti Karttunen : this is also a version of A003188: a(n) 
               = A003188[ n ] - 2^floor_log_2(A003188[ n ]), that is, the corresponding 
               Gray code expansion, but with highest 1-bit turned off. Also a(n) 
               = A003188[ n ] - 2^floor_log_2(n).
%C A038554 Comment from John W. Layman (layman(AT)math.vt.edu): {a(n)} is a self-similar 
               sequence under Kimberling's 'upper-trimming' operation.
%H A038554 T. D. Noe, <a href="b038554.txt">Table of n, a(n) for n=0..4096</a>
%H A038554 C. Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">
               Fractal sequences</a>
%H A038554 J. W. Layman, <a href="http://www.math.vt.edu/people/layman/sequences/
               sequences.htm">View the fractal-like graph of a(n) vs. n</a>
%H A038554 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A038554 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%F A038554 If 2*2^k<=n<3*2^k then a(n)=2^k+a(2^(k+2)-n-1); if 3*2^k<=n<4*2^k then 
               a(n)=a(n-2^(k+1)) - Henry Bottomley (se16(AT)btinternet.com), May 
               11 2000
%F A038554 G.f. 1/(1-x) * sum(k>=0, 2^k(t^4-t^3+t^2)/(1+t^2), t=x^2^k). - Ralf Stephan 
               (ralf(AT)ark.in-berlin.de), Sep 10 2003
%F A038554 a(0)=0, a(2n) = 2a(n) + [n odd], a(2n+1) = 2a(n) + [n>0 even]. - Ralf 
               Stephan (ralf(AT)ark.in-berlin.de), Oct 20 2003
%F A038554 a(0) = a(1) = 0, a(4n) = 2a(2n), a(4n+2) = 2a(2n+1)+1, a(4n+1) = 2a(2n)+1, 
               a(4n+3) = 2a(2n+1). Proof by Nikolaus Meyberg following a conjecture 
               by Ralf Stephan.
%e A038554 If n=18=10010, derivative is (1+0)(0+0)(0+1)(1+0) = 1011, so a(18)=11.
%p A038554 A038554 := proc(n) local i,b,ans; ans := 0; b := convert(n,base,2); for 
               i to nops(b)-1 do ans := ans+((b[ i ]+b[ i+1 ]) mod 2)*2^(i-1); od; 
               RETURN(ans); end; [ seq(A038554(i),i=0..100) ];
%Y A038554 Cf. A038570, A038571. See A003415 for another definition of the derivative 
               of a number.
%Y A038554 Cf. A038556 (rotates n instead of shifting)
%Y A038554 Sequence in context: A137396 A167666 A115352 this_sequence A100329 A081247 
               A144633
%Y A038554 Adjacent sequences: A038551 A038552 A038553 this_sequence A038555 A038556 
               A038557
%K A038554 nonn,nice,easy
%O A038554 0,5
%A A038554 N. J. A. Sloane (njas(AT)research.att.com).
%E A038554 More terms from Erich Friedman (erich.friedman(AT)stetson.edu).