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A094888 Decimal expansion of 2*Pi*phi, where phi = (1+sqrt(5))/2.

%I A094888 #48 Nov 23 2024 05:50:47
%S A094888 1,0,1,6,6,4,0,7,3,8,4,6,3,0,5,1,9,6,3,1,6,1,9,0,1,8,0,2,6,4,8,4,3,9,
%T A094888 7,6,8,3,6,6,3,6,7,8,5,8,6,4,4,2,3,0,8,2,4,0,9,6,4,6,6,5,6,1,8,4,9,9,
%U A094888 9,5,8,2,8,6,9,0,5,3,9,7,2,0,3,7,3,2,1,7,7,2,4,0,7,0,7,8,8,4,3
%N A094888 Decimal expansion of 2*Pi*phi, where phi = (1+sqrt(5))/2.
%H A094888 Ivan Panchenko, <a href="/A094888/b094888.txt">Table of n, a(n) for n = 2..1000</a>
%H A094888 John Baez, <a href="https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/">Pi and the Golden Ratio</a>, Azimuth, Mar 07 2017.
%H A094888 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A094888 From _Peter Bala_, Nov 03 2019: (Start)
%F A094888 Equals 10*Integral_{x >= 0} cosh(4*x)/cosh(5*x) dx = Integral_{x = 0..1} (1 + x^8)/(1 + x^10) dx .
%F A094888 Equals 100*Sum_{n >= 0} (-1)^n*(2*n + 1)/( (10*n + 1)*(10*n + 9) ). (End)
%F A094888 Equals 10 * Product_{k>=2} 2/sqrt(2 + sqrt(2 + ... sqrt(2 + phi)...)), with k nested radicals (Baez, 2017). - _Amiram Eldar_, May 18 2021
%F A094888 Equals Integral_{x>=0} 1/(1 + x^10) dx = (Pi/10) * csc(Pi/10). - _Bernard Schott_, May 15 2022
%F A094888 Equals Gamma(1/10)*Gamma(9/10). - _Andrea Pinos_, Jul 03 2023
%F A094888 Equals 10 * Product_{k >= 1} (10*k)^2/((10*k)^2 - 1). - _Antonio Graciá Llorente_, Mar 15 2024
%F A094888 Equals 10 * Product_{k>=2} (1 + (-1)^k/A090771(k)). - _Amiram Eldar_, Nov 23 2024
%F A094888 Equals 2*A094886 = 10*A135155/e. - _Hugo Pfoertner_, Nov 23 2024
%e A094888 10.16640738463051963161901802648439768366367858644230824...
%p A094888 evalf(Pi*(1+sqrt(5)), 121);  # _Alois P. Heinz_, May 16 2022
%t A094888 RealDigits[2 * Pi * GoldenRatio, 10, 100][[1]] (* _Amiram Eldar_, May 18 2021 *)
%Y A094888 Cf. A000796, A001622, A090771, A094886, A135155.
%Y A094888 Integral_{x>=0} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), A352324 (m=5), A019670 (m=6), A352125 (m=8), this sequence (m=10).
%K A094888 cons,nonn
%O A094888 2,4
%A A094888 _N. J. A. Sloane_, Jun 15 2004