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A037861 (Number of 0's)-(number of 1's) in base 2 representation of n. 36
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)/((2n)(2n+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

J. Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005) 61-65.

J. Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), 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.

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

R. Stephan, Table of generating functions

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Index entries for sequences related to binary expansion of n

FORMULA

a(n) = A023416(n) - A000120(n) = A029837(n) - 2*A000120(n) = 2*A023416(n) - A029837(n); a(2n) = a(n)+1; a(2n+1) = a(2n) - 2 = a(n) - 1. - Henry Bottomley, Oct 27 2000

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

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 A277647

Adjacent sequences:  A037858 A037859 A037860 * A037862 A037863 A037864

KEYWORD

base,sign,look,changed

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified April 20 02:21 EDT 2018. Contains 302772 sequences. (Running on oeis4.)