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A092731 Decimal expansion of Pi^5.

%I A092731 #28 Jun 21 2024 06:36:42
%S A092731 3,0,6,0,1,9,6,8,4,7,8,5,2,8,1,4,5,3,2,6,2,7,4,1,3,1,0,0,4,3,4,3,5,6,
%T A092731 0,6,4,8,0,3,0,0,7,0,6,6,2,8,0,7,4,9,9,0,5,5,3,4,9,2,4,4,3,6,8,6,2,3,
%U A092731 4,9,9,2,1,3,3,6,1,4,0,2,4,4,8,5,7,8,3,5,0,0,4,7,3,5,0,5,1,1,8,9,0,4,0,3,7
%N A092731 Decimal expansion of Pi^5.
%H A092731 G. C. Greubel, <a href="/A092731/b092731.txt">Table of n, a(n) for n = 3..10000</a>
%H A092731 Jean-Christophe Pain, <a href="https://zenodo.org/record/6915559#.YzmuOkrP1hE">The fifth power of Pi: new series representation involving the golden ratio and an application in physics</a>, 2022.
%H A092731 Jean-Christophe Pain, <a href="https://arxiv.org/abs/2208.02624">New series representations for any positive power of Pi from a relation involving trigonometric functions</a>, 2208.02624 [math.NT], 2022.
%H A092731 Kh. Hessami Pilehrood and Tatiana Hessami Pilehrood, <a href="https://doi.org/10.46298/dmtcs.504">Series acceleration formulas for beta values</a>, Discr. Math. Theor. Comp. Sci. 12 (2) (2010) 223-236.
%H A092731 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A092731 From _Peter Bala_, Oct 31 2019: (Start)
%F A092731 Pi^5 = (4!/(2*305)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/6)^5 + 1/(n + 5/6)^5 ), where 305 = ((3^5 + 1)/4)*A000364(2) = A002437(2).
%F A092731 Pi^5 = (4!/(2*3905)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/10)^5 - 1/(n + 3/10)^5 - 1/(n + 7/10)^5 + 1/(n + 9/10)^5 ), where 3905 = ((5^5 - 1)/4)*A000364(2).
%F A092731 Cf. A019692, A091925 and A092735. (End)
%e A092731 306.0196847852814532
%t A092731 RealDigits[Pi^5, 10, 100][[1]] (* _G. C. Greubel_, Mar 09 2018 *)
%o A092731 (PARI) Pi^5 \\ _G. C. Greubel_, Mar 09 2018
%o A092731 (Magma) R:= RealField(100); (Pi(R))^5; // _G. C. Greubel_, Mar 09 2018
%Y A092731 Cf. A000796, A002161, A019692, A091925, A092735, A000364, A002437.
%K A092731 cons,nonn
%O A092731 3,1
%A A092731 _Mohammad K. Azarian_, Apr 12 2004