%I #19 Oct 01 2018 03:30:43
%S 5,6,8,3,0,0,0,0,3,1,4,6,2,3,5,1,7,8,7,6,0,3,3,0,4,1,1,0,3,3,1,7,5,1,
%T 5,1,4,0,7,5,2,6,6,7,4,7,8,2,5,4,0,6,1,2,2,7,2,9,5,6,7,0,5,1,8,7,7,9,
%U 2,0,8,9,7,2,4,5,9,4,0,0,2,8,0,8,2,5,7,1,4,5,4,1,5,5,2,8,5,3,2,2,7,4
%N Decimal expansion of (PolyGamma(1,(1+sqrt(5))/4)-PolyGamma(1,(3+sqrt(5))/4))/2.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 424.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanContinuedFractions.html">Ramanujan Continued Fractions</a>
%e 0.568300003...
%t 4*Integrate[(x*Sech[x])/E^(Sqrt[5]*x), {x, 0, Infinity}]
%t RealDigits[(PolyGamma[1,(1+Sqrt[5])/4]-PolyGamma[1,(3+Sqrt[5])/4])/2, 10, 100][[1]] (* _Vaclav Kotesovec_, Aug 16 2015 *)
%o (PARI)
%o polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
%o (polygamma(1, (1+sqrt(5))/4) - polygamma(1, (3+sqrt(5))/4))/2 \\ _Gheorghe Coserea_, Sep 30 2018
%Y Cf. A091660.
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Jan 26 2004
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