A164916 Denominators of a BBP series for Pi/4.

1, -8, -20, -24, 144, -384, -832, -896, 4352, -10240, -21504, -22528, 102400, -229376, -475136, -491520, 2162688, -4718592, -9699328, -9961472, 42991616, -92274688, -188743680, -192937984, 822083584, -1744830464, -3556769792

Offset: 0

Author

Jaume Oliver Lafont, Aug 31 2009

Keywords

• frac
• sign

Comments

From the BBP formula for Pi, the following expression for Pi/4 in unit numerators is obtained

Pi/4 = Sum((1/(8k+1)+1/(-2*(8k+4))+1/(-4*(8k+5))+1/(-4*(8k+6)))/16^k, k>=0)

Therefore a(n) such that

a(4*n) = (8*n+1)*16^n.

a(4*n+1) = -2*(8*n+4)*16^n.

a(4*n+2) = -4*(8*n+5)*16^n.

a(4*n+3) = -4*(8*n+6)*16^n.

has

Sum_{n >= 0} (1/a(n)) = Pi/4.

Using PARI/GP suminf(n=0,1/(2^(n-2)*(2*(-1+(-1)^n+(1-I)*(-I)^n+(1+I)*I^n)+(-3+3*(-1)^n+(4-I)*(-I)^n+(4+I)*I^n)*n)))= 0.7853981633974483096156608454...=Pi/4. - Alexander R. Povolotsky, Sep 01 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A048581, A066968, A154925, A154962, A156269, A157142.

Programs

Mathematica

CoefficientList[Series[(1 - 8*x - 20*x^2 - 24*x^3 + 112*x^4 - 128*x^5 - 192*x^6 - 128*x^7)/(1 - 16*x^4)^2, {x,0,50}], x] (* G. C. Greubel, Feb 25 2017 *)

PARI

x='x + O('x^50); Vec((1 - 8*x - 20*x^2 - 24*x^3 + 112*x^4 - 128*x^5 - 192*x^6 - 128*x^7)/(1 - 16*x^4)^2) \\ G. C. Greubel, Feb 25 2017

Formula

G.f.: (1-8*x-20*x^2-24*x^3+112*x^4-128*x^5-192*x^6-128*x^7)/(1-16*x^4)^2.

a(n)= 2^(n-2)*(2*(-1+(-1)^n+(1-I)*(-I)^n+(1+I)*I^n)+(-3+3*(-1)^n+(4-I)*(-I)^n+(4+I)*I^n)*n). - Alexander R. Povolotsky, Sep 01 2009

Extensions

Comment section corrected by Jaume Oliver Lafont, Sep 03 2009