A157332 Denominators of Egyptian fraction for Pi/16 based on Machin's formula.

5, -956, -375, 163823028, 15625, -15596225303980, -546875, 1247220779824098212, 17578125, -91597497639855832244124, -537109375, 6394838587727583881086964116, 15869140625, -431694043145875922302762745864588, -457763671875

Offset: 0

Author

Jaume Oliver Lafont, Feb 27 2009

Keywords

• frac
• sign
• easy

Comments

Machin's formula: Pi/4 = 4*atan(1/5) - atan(1/239).

Sum_{n>=0} 1/a(n) = Pi/16 = atan(1/5) - (1/4)*atan(1/239).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..415
• X. Gourdon and P. Sebah, Collection of series for Pi

Crossrefs

Cf. A019683, A072172, A072173.

Programs

GAP

a:= function(n)
    if n mod 2=0 then return (-1)^(n/2)*(n+1)*5^(n+1);
    else return -4*(-1)^((n-1)/2)*n*(239)^n;
    fi;
  end;
List([0..15], n-> a(n) ); # G. C. Greubel, Aug 26 2019

Magma

R:=PowerSeriesRing(Integers(), 15); Coefficients(R!( 5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2 )); // G. C. Greubel, Aug 26 2019

Maple

seq(coeff(series(5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2, x, n+1), x, n), n = 0..15); # G. C. Greubel, Aug 26 2019

Mathematica

CoefficientList[Series[5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2, {x,0,15}], x] (* G. C. Greubel, Aug 26 2019 *)

PARI

my(x='x+O('x^15)); Vec(5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2) \\ G. C. Greubel, Aug 26 2019

Sage

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2 ).list()
A077952_list(15) # G. C. Greubel, Aug 26 2019

Formula

a(2n) = (2*n+1)*5^(2*n+1)*(-1)^n,

a(2n+1) = -4*(2*n+1)*239^(2*n+1)*(-1)^n.

G.f.: 5*(1-25*x^2)/(1+25*x^2)^2 - 956*x*(1-57121*x^2)/(1+57121*x^2)^2

Extensions

More terms from Colin Barker, Aug 07 2013