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

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, 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, 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

Offset

2,4

Links

Ivan Panchenko, Table of n, a(n) for n = 2..1000

John Baez, Pi and the Golden Ratio, Azimuth, Mar 07 2017.

Index entries for transcendental numbers.

Formula

From Peter Bala, Nov 03 2019: (Start)

Equals 10*Integral_{x >= 0} cosh(4*x)/cosh(5*x) dx = Integral_{x = 0..1} (1 + x^8)/(1 + x^10) dx .

Equals 100*Sum_{n >= 0} (-1)^n*(2*n + 1)/( (10*n + 1)*(10*n + 9) ). (End)

Equals 10 * Product_{k>=2} 2/sqrt(2 + sqrt(2 + ... sqrt(2 + phi)...)), with k nested radicals (Baez, 2017). - Amiram Eldar, May 18 2021

Equals Integral_{x>=0} 1/(1 + x^10) dx = (Pi/10) * csc(Pi/10). - Bernard Schott, May 15 2022

Equals Gamma(1/10)*Gamma(9/10). - Andrea Pinos, Jul 03 2023

Equals 10 * Product_{k >= 1} (10*k)^2/((10*k)^2 - 1). - Antonio Graciá Llorente, Mar 15 2024

Equals 10 * Product_{k>=2} (1 + (-1)^k/A090771(k)). - Amiram Eldar, Nov 23 2024

Equals 2*A094886 = 10*A135155/e. - Hugo Pfoertner, Nov 23 2024

Example

10.16640738463051963161901802648439768366367858644230824...

Maple

evalf(Pi*(1+sqrt(5)), 121);  # Alois P. Heinz, May 16 2022

Mathematica

RealDigits[2 * Pi * GoldenRatio, 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

Crossrefs

Cf. A000796, A001622, A090771, A094886, A135155.

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).

Sequence in context: A143937 A019133 A214581 * A159702 A255188 A164510

Adjacent sequences: A094885 A094886 A094887 * A094889 A094890 A094891

Keyword

cons,nonn,changed

Author

N. J. A. Sloane, Jun 15 2004

Status

approved