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A038554 Derivative of n: write n in binary, replace each pair of adjacent bits with their mod 2 sum (a(0)=a(1)=0 by convention). Also n XOR (n shift 1). 19
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, 16, 17, 19, 18, 22, 23, 21, 20, 28, 29, 31, 30, 26, 27, 25, 24, 8, 9, 11, 10, 14, 15, 13, 12, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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).

From John W. Layman: {a(n)} is a self-similar sequence under Kimberling's 'upper-trimming' operation.

a(A000225(n)) = 0; a(A062289(n)) > 0; a(A038558(n)) = n. - Reinhard Zumkeller, Mar 06 2013

LINKS

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

FORMULA

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, 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, Sep 10 2003

a(0)=0, a(2n) = 2a(n) + [n odd], a(2n+1) = 2a(n) + [n>0 even]. - Ralf Stephan, 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.

EXAMPLE

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

MAPLE

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) ];

MATHEMATICA

a[0] = a[1] = 0; a[n_ /; Mod[n, 4] == 0] := a[n] = 2*a[n/2]; a[n_ /; Mod[n, 4] == 1] := a[n] =  2*a[(n-1)/2] + 1; a[n_ /; Mod[n, 4] == 2] := a[n] = 2*a[n/2] + 1; a[n_ /; Mod[n, 4] == 3] := a[n] = 2*a[(n-1)/2]; Table[a[n], {n, 0, 81}] (* Jean-Fran├žois Alcover, Jul 13 2012, after Ralf Stephan *)

Table[FromDigits[Mod[Total[#], 2]&/@Partition[IntegerDigits[n, 2], 2, 1], 2], {n, 0, 90}] (* Harvey P. Dale, Oct 27 2015 *)

PROG

(Haskell)

import Data.Bits (xor)

a038554 n = foldr (\d v -> v * 2 + d) 0 $ zipWith xor bs $ tail bs

   where bs = a030308_row n

-- Reinhard Zumkeller, May 26 2013, Mar 06 2013

(PARI) a003188(n)=bitxor(n, n>>1);

a(n)=if(n<2, 0, a003188(n) - 2^logint(a003188(n), 2)); \\ Indranil Ghosh, Apr 26 2017

(Python)

import math

def a003188(n): return n^(n>>1)

def a(n): return 0 if n<2 else a003188(n) - 2**int(math.floor(math.log(a003188(n), 2))) # Indranil Ghosh, Apr 26 2017

CROSSREFS

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

Cf. A038556 (rotates n instead of shifting)

Cf. A000035.

Cf. A030308.

Sequence in context: A167666 A115352 A275808 * A100329 A193535 A081247

Adjacent sequences:  A038551 A038552 A038553 * A038555 A038556 A038557

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Erich Friedman

STATUS

approved

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Last modified May 24 17:28 EDT 2017. Contains 286997 sequences.