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A037861
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(Number of 0's)-(number of 1's) in base 2 representation of n.
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23
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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; internal format)
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OFFSET
| 0,4
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COMMENTS
| -sum(n>=1, a(n)/((2n)(2n+1))) = the "alternating Euler constant" log(4/Pi) = 0.24156... - (see A094640 and Sondow 2005, 2010).
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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
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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 (se16(AT)btinternet.com), Oct 27 2000
G.f. satisfies A(x) = (1+x)A(x^2) - x(2+x)/(1+x) - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 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 (reinhard.zumkeller(AT)gmail.com), Dec 31 2007
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MATHEMATICA
| Table[ Abs[ Count[ IntegerDigits[n, 2], 0] - Count[ IntegerDigits[n, 2], 1] ], {n, 0, 75} ]
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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.
Sequence in context: A090379 A077254 A074761 * A145037 A158052 A158378
Adjacent sequences: A037858 A037859 A037860 * A037862 A037863 A037864
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KEYWORD
| base,sign
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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