A104895 a(0)=0; thereafter a(2n) = -2*a(n), a(2n+1) = 2*a(n) - 1.

0, -1, -2, 1, -4, 3, 2, -3, -8, 7, 6, -7, 4, -5, -6, 5, -16, 15, 14, -15, 12, -13, -14, 13, 8, -9, -10, 9, -12, 11, 10, -11, -32, 31, 30, -31, 28, -29, -30, 29, 24, -25, -26, 25, -28, 27, 26, -27, 16, -17, -18, 17, -20, 19, 18, -19, -24, 23, 22, -23, 20, -21, -22, 21, -64, 63, 62, -63, 60, -61, -62, 61, 56, -57, -58, 57, -60, 59

Offset: 0

Author

Philippe Deléham, Apr 24 2005

Keywords

• sign
• look
• nice

Comments

Columns of table in A104894 written in base 10.

Conjecture: the positions where 0, 1, 2, 3, ... appear are given by A048724; the positions where -1, -2, -3, ... appear are given by A065621.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A048724, A065621, A104894.

The negative of entry A065620.

Programs

Haskell

import Data.List (transpose)
a104895 n = a104895_list !! n
a104895_list = 0 : concat (transpose [map (negate . (+ 1)) zs, tail zs])
               where zs = map (* 2) a104895_list
-- Reinhard Zumkeller, Mar 26 2014

Maple

f:=proc(n) option remember; if n=0 then RETURN(0); fi; if n mod 2 = 0 then RETURN(2*f(n/2)); else RETURN(-2*f((n-1)/2)-1); fi; end;

Mathematica

a[0] = 0;
a[n_]:= a[n]= If[EvenQ[n], 2 a[n/2], -2 a[(n-1)/2] - 1];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 03 2018 *)

Sage

def a(n):
    if (n==0): return 0
    elif (mod(n,2)==0): return 2*a(n/2)
    else: return -2*a((n-1)/2) - 1
[a(n) for n in (0..100)] # G. C. Greubel, Jun 15 2021

Formula

a(0) = 0 and for k>=0, 0<= j <2^k, a(2^k + j) = a(j) + 2^k if a(j)<0, a(2^k + j) = a(j) - 2^k if a(j)>=0.

Sum_{0 <= n <= 2^k - 1} a(n) = - 2^(k-1).

Sum_{0 <= n <= 2^k - 1} |a(n)| = 4^(k-1).

a(n) = -A065620(n). - M. F. Hasler, Apr 16 2018

Extensions

Corrected by N. J. A. Sloane, Nov 05 2005

Edited by N. J. A. Sloane, Apr 25 2018