0,5

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).

Comment from John W. Layman (layman(AT)math.vt.edu): {a(n)} is a self-similar sequence under Kimberling's 'upper-trimming' operation.

T. D. Noe, Table of n, a(n) for n=0..4096

C. Kimberling, Fractal sequences

J. W. Layman, View the fractal-like graph of a(n) vs. n

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

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

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

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

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.

If n=18=10010, derivative is (1+0)(0+0)(0+1)(1+0) = 1011, so a(18)=11.

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) ];

Cf. A038570, A038571. See A003415 for another definition of the derivative of a number.

Cf. A038556 (rotates n instead of shifting)

Sequence in context: A137396 A167666 A115352 this_sequence A100329 A081247 A144633

Adjacent sequences: A038551 A038552 A038553 this_sequence A038555 A038556 A038557

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Erich Friedman (erich.friedman(AT)stetson.edu).