A244928 - OEIS

%I #15 May 13 2024 21:05:12

%S 2,6,5,8,6,4,9,5,8,2,7,9,3,0,6,9,8,2,6,9,1,8,7,5,0,8,6,3,9,7,1,2,0,6,

%T 8,7,6,4,2,7,8,3,8,2,3,9,7,5,1,3,8,9,9,9,3,8,0,5,9,7,4,1,5,3,2,8,5,7,

%U 4,3,9,5,1,3,0,2,7,7,1,1,4,0,5,4,4,1,1,4,0,7,0,3,2,0,5,7,7,1,7,4,0,4,5,7,1

%N Decimal expansion of Ti_2(2-sqrt(3)), where Ti_2 is the inverse tangent integral function.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.7.6 Inverse Tangent Integral, p. 57.

%H G. C. Greubel, <a href="/A244928/b244928.txt">Table of n, a(n) for n = 0..10000</a>

%H E. D. Krupnikov, K. S. Kölbig, <a href="https://doi.org/10.1016/S0377-0427(96)00111-2">Some special cases of the generalized hypergeometric function (q+1)Fq</a>, J. Comp. Appl. Math. 78 (1997) 79-95.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/InverseTangentIntegral.html">Inverse Tangent Integral</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>

%F 2/3*G + Pi/12*log(2-Sqrt(3)), where G is Catalan's number.

%F Also equals i/2*(polylog(2, -i*(2-sqrt(3))) - polylog(2, i*(2-sqrt(3)))), with i = sqrt(-1).

%F Equals 3F2(1/2,1,1; 3/2,3/2 ; 1/4)/4 [Krupnikov] - _R. J. Mathar_, May 13 2024

%e 0.26586495827930698269187508639712068764278382397513899938059741532857439513...

%t 2/3*Catalan + Pi/12*Log[2 - Sqrt[3]] // RealDigits[#, 10, 105]& // First

%o (PARI) default(realprecision, 100); (2/3)*Catalan + Pi/12*log(2 - sqrt(3)) \\ _G. C. Greubel_, Aug 25 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); (2/3)*Catalan(R) + Pi(R)/12*Log(2 - Sqrt(3)); // _G. C. Greubel_, Aug 25 2018

%Y Cf. A006752, A244929.

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Jul 08 2014