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A352324 Decimal expansion of 4*Pi / (5*sqrt(10-2*sqrt(5))).

%I A352324 #37 Nov 22 2024 11:05:50
%S A352324 1,0,6,8,9,5,9,3,3,2,1,1,5,5,9,5,1,1,3,4,2,5,1,8,4,3,7,2,5,0,6,8,8,2,
%T A352324 6,3,9,9,0,1,4,5,0,9,2,5,2,6,6,5,2,4,5,8,6,0,0,6,6,6,3,2,5,6,3,7,9,6,
%U A352324 2,1,1,4,9,6,7,9,0,7,4,9,1,3,2,2,7,8,0,3,8,7,7,9,4
%N A352324 Decimal expansion of 4*Pi / (5*sqrt(10-2*sqrt(5))).
%C A352324 Cauchy's residue theorem implies that Integral_{x=0..oo} 1/(1 + x^m) dx = (Pi/m) * csc(Pi/m); this is the case m = 5.
%C A352324 The area of a circle circumscribing a unit-area regular decagon.
%D A352324 Jean-François Pabion, Éléments d'Analyse Complexe, licence de Mathématiques, page 111, Ellipses, 1995.
%H A352324 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A352324 Equals Integral_{x=0..oo} 1/(1 + x^5) dx.
%F A352324 Equals (Pi/5) *csc(Pi/5).
%F A352324 Equals (1/2) * A019694 * A121570.
%F A352324 Equals 1/Product_{k>=1} (1 - 1/(5*k)^2). - _Amiram Eldar_, Mar 12 2022
%F A352324 Equals Product_{k>=2} (1 + (-1)^k/A047209(k)). - _Amiram Eldar_, Nov 22 2024
%F A352324 Equals 1/A371604 = A377405/5. - _Hugo Pfoertner_, Nov 22 2024
%e A352324 1.0689593321155951134251843725068826399014509252665...
%p A352324 evalf(4*Pi / (5*(sqrt(10-2sqrt(5)))), 100);
%t A352324 First[RealDigits[N[4Pi/(5Sqrt[10-2Sqrt[5]]), 93]]] (* _Stefano Spezia_, Mar 12 2022 *)
%Y A352324 Cf. A019694, A047209, A121570, A371604, A377405.
%Y A352324 Integral_{x=0..oo} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), this sequence (m=5), A019670 (m=6), A352125 (m=8), A094888 (m=10).
%K A352324 nonn,cons
%O A352324 1,3
%A A352324 _Bernard Schott_, Mar 12 2022