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A156269 Denominators of a series expansion for Pi/2.

%I A156269 #13 Dec 16 2016 16:52:54
%S A156269 1,2,6,-20,-24,-56,144,160,352,-832,-896,-1920,4352,4608,9728,-21504,
%T A156269 -22528,-47104,102400,106496,221184,-475136,-491520,-1015808,2162688,
%U A156269 2228224,4587520,-9699328,-9961472,-20447232,42991616,44040192,90177536
%N A156269 Denominators of a series expansion for Pi/2.
%C A156269 Numerators are all 1.
%C A156269 Sum_{n >= 0} 1/a(n) = Pi/2.
%C A156269 This sequence is based on Adamchik and Wagon's BBP-type three-term formula for Pi, namely Pi = Sum_{n >= 0} (-1/4)^n*(2/(4*n + 1) + 2/(4*n + 2) + 1/(4*n + 3)).
%C A156269 From _Peter Bala_, Jun 16 2016: (Start)
%C A156269 The reciprocals 1/a(n) appear as coefficients in the Maclaurin series for 2*arctan(z/(2 - z)) = z + z^2/2 + z^3/6 - z^5/20 - z^6/24 - z^7/56 + ... (the radius of convergence is sqrt(2)).
%C A156269 Setting z = 1 gives Pi/2 = Sum_{n >= 0} 1/a(n) as observed above. Setting z = 2 - sqrt(2) gives a series for Pi/4 in terms of a(n). Setting z = +- sqrt(2), and using Abel's theorem on power series, gives two further series for Pi involving a(n). (End)
%H A156269 V. Adamchik and S. Wagon, <a href="http://www.cs.cmu.edu/~adamchik/articles/pi/pi.htm">Pi: A 2000-Year Search Changes Direction</a>
%H A156269 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, -8, 0, 0, -16).
%F A156269 G.f.: (1+2*x+6*x^2-12*x^3-8*x^4-8*x^5)/(1+4*x^3)^2.
%F A156269 From _Peter Bala_, Jun 16 2016: (Start)
%F A156269 a(3*n) = (-4)^n*(4*n + 1);
%F A156269 a(3*n + 1) = (-4)^n*(4*n + 2);
%F A156269 a(3*n + 2) = (-4)^n*(8*n + 6). (End)
%p A156269 A156269 := n -> if `mod`(n, 3) = 0 then (-4)^(n/3)*(4*n/3 + 1) elif `mod`(n, 3) = 1 then (-4)^((n-1)/3)*(4*(n-1)/3 + 2) else (-4)^((n-2)/3)*(8*(n-2)/3 + 6) end if:
%p A156269 seq(A156269(n), n = 1 .. 35); # _Peter Bala_, Jun 16 2016
%t A156269 CoefficientList[Series[(1+2x+6x^2-12x^3-8x^4-8x^5)/(1+4x^3)^2,{x,0,40}],x] (* or *) LinearRecurrence[{0,0,-8,0,0,-16},{1,2,6,-20,-24,-56},40] (* _Harvey P. Dale_, Dec 16 2016 *)
%Y A156269 Cf. A000796, A154925, A154962.
%K A156269 sign,easy
%O A156269 0,2
%A A156269 _Jaume Oliver Lafont_, Feb 07 2009, Feb 21 2009