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A175578 Decimal expansion of the sum over the inverse icosahedral numbers.

%I A175578 #16 Aug 29 2018 16:47:24
%S A175578 1,1,2,3,5,6,5,9,6,6,8,9,9,2,5,1,8,8,7,5,7,3,9,3,7,5,7,9,0,1,5,8,7,9,
%T A175578 6,4,5,3,5,3,8,1,1,4,1,6,4,8,5,5,0,4,9,8,0,6,0,6,1,7,0,2,6,9,2,9,8,1,
%U A175578 9,2,6,0,3,3,6,1,5,4,2,6,6,9,5,8,2,6,0,9,2,1,0,6,8,8,8,7,7,8,1,0,7,2,6,4,7
%N A175578 Decimal expansion of the sum over the inverse icosahedral numbers.
%C A175578 Defined by sum_{n>=1} 1/A006564(n) = 1/1 + 1/12 +1/48 + 1/124 +...
%C A175578 Equals gamma + Pi*sqrt(5/3)*tanh(Pi*sqrt(15)/10)/2 + Re psi( 1/2+i*sqrt(15)/10 ), where psi is the digamma function, i the imaginary unit, Pi = A000796, sqrt(15)=A010472, gamma=A001620.
%H A175578 G. C. Greubel, <a href="/A175578/b175578.txt">Table of n, a(n) for n = 1..10000</a>
%e A175578 1.12356596689925188757393..
%p A175578 Digits := 120 : gamma+ Psi(1/2+sqrt(15)*I/10)+sqrt(15)/6*Pi*tanh(Pi*sqrt(15)/10) ; evalf(Re(%)) ;
%t A175578 N[EulerGamma + PolyGamma[1/2 + (I*Sqrt[15])/10] + (1/2)*Tanh[(Pi*Sqrt[15])/10]*Pi*Sqrt[5/3] // Re, 105] // RealDigits // First (* _Jean-François Alcover_, Feb 05 2013 *)
%o A175578 (PARI) Euler+Pi*sqrt(5/3)*tanh(Pi*sqrt(15)/10)/2+real(psi(1/2+ I*sqrt(15)/10)) \\ _Charles R Greathouse IV_, Jul 19 2013
%Y A175578 Cf. A006564 (icosahedral numbers).
%Y A175578 Cf. sums of inverses: A152623 (tetrahedral numbers), A002117 (cubes), A175577 (octahedral numbers), A295421 (dodecahedral numbers).
%K A175578 cons,easy,nonn
%O A175578 1,3
%A A175578 _R. J. Mathar_, Jul 15 2010