(MAGMAMagma) R<x>:=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

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

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

G. C. Greubel, <a href="/A157332/b157332.txt">Table of n, a(n) for n = 0..415</a>

a(2n) = (2n2*n+1)*5^(2n+1)2*(-1)^n, a(2n+1)=-4*(2n+1)239^(2n+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

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

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

(MAGMA) R<x>:=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

(Sage)

def A077952_list(prec):

P.<x> = 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

(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

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frac,sign,changed,easy

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

More terms from Colin Barker, Aug 07 2013

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editing

_Jaume Oliver Lafont (joliverlafont(AT)gmail.com), _, Feb 27 2009

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

0,1

Machin's formula: Pi/4 = 4atan(1/5) - atan(1/239). Sum_{n>=0} 1/a(n) = Pi/16 = atan(1/5) - (1/4)atan(1/239).

X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Pi/piSeries.html">Collection of series for Pi</a>

a(2n)=(2n+1)*5^(2n+1)*(-1)^n, a(2n+1)=-4*(2n+1)239^(2n+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

Cf. A019683, A072172, A072173.

frac,sign

Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 27 2009

approved