A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

1, -1, 0, -2, 1, -1, -1, -3, 2, 0, 0, -2, 0, -2, -2, -4, 3, 1, 1, -1, 1, -1, -1, -3, 1, -1, -1, -3, -1, -3, -3, -5, 4, 2, 2, 0, 2, 0, 0, -2, 2, 0, 0, -2, 0, -2, -2, -4, 2, 0, 0, -2, 0, -2, -2, -4, 0, -2, -2, -4, -2, -4, -4, -6, 5, 3, 3, 1, 3, 1, 1, -1, 3

Offset: 0

Author

Clark Kimberling

Keywords

• base
• sign
• look
• easy

Comments

-Sum_{n>=1} a(n)/((2*n)*(2*n+1)) = the "alternating Euler constant" log(4/Pi) = 0.24156... - (see A094640 and Sondow 2005, 2010).

a(A072600(n)) < 0; a(A072601(n)) <= 0; a(A031443(n)) = 0; a(A072602(n)) >= 0; a(A072603(n)) > 0; a(A031444(n)) = 1; a(A031448(n)) = -1; abs(a(A089648(n))) <= 1. - Reinhard Zumkeller, Feb 07 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.
• Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
• Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
• Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
• Ralf Stephan, Table of generating functions (ps file).
• Ralf Stephan, Table of generating functions (pdf file).
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A031443 for n when a(n)=0, A053738 for n when a(n) odd, A053754 for n when a(n) even, A030300 for a(n+1) mod 2.

Cf. A066879, A094640, A110625, A145037.

Cf. A072600, A072601, A072602, A072603, A031444, A031448, A089648.

See A268289 for a recurrence based on this sequence.

Programs

Haskell

a037861 n = a023416 n - a000120 n  -- Reinhard Zumkeller, Aug 01 2013

Maple

A037861:= proc(n) local L;
     L:= convert(n,base,2);
     numboccur(0,L) - numboccur(1,L)
end proc:
map(A037861, [$0..100]); # Robert Israel, Mar 08 2016

Mathematica

Table[Count[ IntegerDigits[n, 2], 0] - Count[IntegerDigits[n, 2], 1], {n, 0, 75}]

PARI

a(n) = if (n==0, 1, 1 + logint(n, 2) - 2*hammingweight(n)); \\ Michel Marcus, May 15 2020 and Jun 16 2020

Python

def A037861(n):
    return 2*format(n,'b').count('0')-len(format(n,'b')) # Chai Wah Wu, Mar 07 2016

Formula

From Henry Bottomley, Oct 27 2000: (Start)

a(n) = A023416(n) - A000120(n) = A029837(n) - 2*A000120(n) = 2*A023416(n) - A029837(n).

a(2*n) = a(n) + 1; a(2*n + 1) = a(2*n) - 2 = a(n) - 1. (End)

G.f. satisfies A(x) = (1 + x)*A(x^2) - x*(2 + x)/(1 + x). - Franklin T. Adams-Watters, Dec 26 2006

a(n) = b(n) for n > 0 with b(0) = 0 and b(n) = b(floor(n/2)) + (-1)^(n mod 2). - Reinhard Zumkeller, Dec 31 2007

G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(2^k)*(x^(2^k) - 1)/(1 + x^(2^k)). - Ilya Gutkovskiy, Apr 07 2018