A212002 - OEIS

A212002

Decimal expansion of (2*Pi)^2.

3, 9, 4, 7, 8, 4, 1, 7, 6, 0, 4, 3, 5, 7, 4, 3, 4, 4, 7, 5, 3, 3, 7, 9, 6, 3, 9, 9, 9, 5, 0, 4, 6, 0, 4, 5, 4, 1, 2, 5, 4, 7, 9, 7, 6, 2, 8, 9, 6, 3, 1, 6, 2, 5, 0, 5, 6, 5, 3, 3, 9, 7, 5, 0, 4, 8, 8, 0, 1, 7, 9, 2, 8, 9, 6, 7, 6, 8, 2, 0, 9, 7, 2, 0, 0, 7

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OFFSET

2,1

COMMENTS

This constant appears in Kepler's 3rd Law, T^2 = (2*Pi)^2/GM*a^3 where a is the semi-major axis of a planet orbiting the Sun, T is its period, and GM is the standard gravitational parameter. - Raphie Frank, Dec 13 2012

García & Marco give a generalized zeta regularization by which this is the value of the product of the primes. - Charles R Greathouse IV, Jun 17 2013

LINKS

Table of n, a(n) for n=2..87.

J.-P. Allouche, The zeta-regularized product of odious numbers, arXiv:1906.10532 [math.NT], 2019.

Hyperphysics, Kepler's Laws

E. Muñoz García and R. Pérez Marco, The product over all primes is 4Pi^2, Communications in Mathematical Physics, Vol. 277, No. 1 (2008), pp. 69-81.

Hengguang Li and Jeffrey S. Ovall, A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential, Discrete and Continuous Dynamical Systems Series B, Volume 20, Number 5, July 2015, pp. 1377-1391. Also doi:10.3934/dcdsb.2015.20.1377

Wikipedia, Pendulum.

Index entries for transcendental numbers

FORMULA

Equals Product_{k=1..10, gcd(k,10)==1} Gamma(k/10) = Gamma(1/10)*Gamma(3/10)*Gamma(7/10)*Gamma(9/10). - Amiram Eldar, Jun 12 2021

Equals lim_{n->oo} |B(2*n)/B(2*n+2)|*(2*n+1)*(2*n+2), where B(n) denotes the n-th Bernoulli number. - Peter Luschny, Dec 09 2021

EXAMPLE

39.4784176043574344753379639995046045412547976289631...

MATHEMATICA

RealDigits[(2*Pi)^2, 10, 120][[1]] (* Harvey P. Dale, Mar 27 2019 *)

PROG

(PARI) 4*Pi^2 \\ Charles R Greathouse IV, Jun 17 2013

CROSSREFS