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A387206 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(6*k-5)^6 + (-1)^(k+1)/(6*k-1)^6.

%I A387206 #16 Aug 26 2025 16:38:28
%S A387206 1,0,0,0,0,5,5,1,6,0,7,9,4,8,6,9,4,6,3,4,7,3,7,5,6,6,1,8,8,4,4,9,2,5,
%T A387206 6,2,5,8,7,8,1,9,7,6,9,3,6,5,2,0,6,5,1,8,5,6,3,1,0,1,8,2,5,7,0,6,1,3,
%U A387206 0,3,5,7,9,3,8,0,0,9,9,7,1,9,2,0,7,8,2,1,6,6,6,2,7,3,2,5,0,6,8,9
%N A387206 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(6*k-5)^6 + (-1)^(k+1)/(6*k-1)^6.
%H A387206 Jason Bard, <a href="/A387206/b387206.txt">Table of n, a(n) for n = 1..1000</a>
%H A387206 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See p. 21. (Misprint contains erroneous minus sign)
%F A387206 Equals (1/358318080) * (PolyGamma(5, 1/12) + PolyGamma(5, 5/12) - PolyGamma(5, 7/12) - PolyGamma(5, 11/12)).
%F A387206 Equals (73/35831808) * (PolyGamma(5, 1/4) - PolyGamma(5, 3/4)). - _Amiram Eldar_, Aug 22 2025
%e A387206 1.000055160794869463473756618844925625878197693652065...
%p A387206 c:= Re(sum((-1)^(k+1)/(6*k-5)^6+(-1)^(k+1)/(6*k-1)^6, k=1..infinity)):
%p A387206 evalf(c, 140);  # _Alois P. Heinz_, Aug 21 2025
%t A387206 RealDigits[(1/358318080)*(PolyGamma[5, 1/12] + PolyGamma[5, 5/12] - PolyGamma[5, 7/12] - PolyGamma[5, 11/12]), 10, 100][[1]]
%o A387206 (PARI) (365/1492992) * (zetahurwitz(6, 1/4) - zetahurwitz(6, 3/4))
%K A387206 nonn,cons,new
%O A387206 1,6
%A A387206 _Jason Bard_, Aug 21 2025