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A037861 (Number of 0's) - (number of 1's) in the base 2 representation of n. 37
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

R. 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

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

MAPLE

f:= proc(n) local L;

L:= convert(n, base, 2);

numboccur(0, L) - numboccur(1, L)

end proc:

map(f, [$0..100]); # Robert Israel, Mar 08 2016

MATHEMATICA

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

PROG

(Haskell)

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

(Python)

def A037861(n):

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

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

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, A031443, A031444, A031448, A089648.

See A268289 for a recurrence based on this sequence.

Sequence in context: A254655 A077254 A074761 * A145037 A267115 A328919

Adjacent sequences:  A037858 A037859 A037860 * A037862 A037863 A037864

KEYWORD

base,sign,look

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified July 2 02:41 EDT 2020. Contains 335389 sequences. (Running on oeis4.)