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

%I A157332 #12 Jul 22 2025 06:22:40
%S A157332 5,-956,-375,163823028,15625,-15596225303980,-546875,
%T A157332 1247220779824098212,17578125,-91597497639855832244124,-537109375,
%U A157332 6394838587727583881086964116,15869140625,-431694043145875922302762745864588,-457763671875
%N A157332 Denominators of Egyptian fraction for Pi/16 based on Machin's formula.
%C A157332 Machin's formula: Pi/4 = 4*atan(1/5) - atan(1/239).
%C A157332 Sum_{n>=0} 1/a(n) = Pi/16 = atan(1/5) - (1/4)*atan(1/239).
%H A157332 G. C. Greubel, <a href="/A157332/b157332.txt">Table of n, a(n) for n = 0..415</a>
%H A157332 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Pi/piSeries.html">Collection of series for Pi</a>
%F A157332 a(2n) = (2*n+1)*5^(2*n+1)*(-1)^n,
%F A157332 a(2n+1) = -4*(2*n+1)*239^(2*n+1)*(-1)^n.
%F A157332 G.f.: 5*(1-25*x^2)/(1+25*x^2)^2 - 956*x*(1-57121*x^2)/(1+57121*x^2)^2
%p A157332 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
%t A157332 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 *)
%o A157332 (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
%o A157332 (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
%o A157332 (Sage)
%o A157332 def A077952_list(prec):
%o A157332     P.<x> = PowerSeriesRing(ZZ, prec)
%o A157332     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()
%o A157332 A077952_list(15) # _G. C. Greubel_, Aug 26 2019
%o A157332 (GAP)
%o A157332 a:= function(n)
%o A157332     if n mod 2=0 then return (-1)^(n/2)*(n+1)*5^(n+1);
%o A157332     else return -4*(-1)^((n-1)/2)*n*(239)^n;
%o A157332     fi;
%o A157332   end;
%o A157332 List([0..15], n-> a(n) ); # _G. C. Greubel_, Aug 26 2019
%Y A157332 Cf. A019683, A072172, A072173.
%K A157332 frac,sign,easy
%O A157332 0,1
%A A157332 _Jaume Oliver Lafont_, Feb 27 2009
%E A157332 More terms from _Colin Barker_, Aug 07 2013