%I #23 Feb 08 2023 16:34:37

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

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

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

%N Decimal expansion of the sum of the reciprocals of the octahedral numbers (A005900).

%C Defined by Sum_{n>=1} 1/A005900(n) = 1/1 + 1/6 + 1/19 + 1/44 + ...

%C Equals 3*(gamma + Re psi(i/sqrt 2) ) = 3* Re(A001620 + psi(i*A010503)) where psi(i*A010503) = -0.1511539... + i*2.3152942... is a digamma function and i the imaginary unit.

%H G. C. Greubel, <a href="/A175577/b175577.txt">Table of n, a(n) for n = 1..10000</a>

%e 1.2781851590909461795403909483...

%p Digits := 120 : 3*(gamma+Psi(I/sqrt(2))); evalf(Re(%)) ;

%t RealDigits[ 3/2*(2*EulerGamma + Re[PolyGamma[0, 1 - I/Sqrt[2]] + PolyGamma[0, 1 + I/Sqrt[2]]]), 10, 105] // First (* _Jean-François Alcover_, Feb 11 2013 *)

%o (PARI) 3*Euler+3*real(psi(I/sqrt(2))) \\ _Charles R Greathouse IV_, Jul 19 2013

%o (PARI) sumnumrat(3/n/(2*n^2 + 1),1) \\ _Charles R Greathouse IV_, Feb 08 2023

%Y Cf. A005900 (octahedral numbers).

%Y Cf. sums of inverses: A152623 (tetrahedral numbers), A002117 (cubes), A295421 (dodecahedral numbers), A175578 (icosahedral numbers).

%K cons,easy,nonn

%O 1,2

%A _R. J. Mathar_, Jul 15 2010