A345738 Decimal expansion of (2*G+1)/Pi, where G is Catalan's constant (A006752).

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

Offset

0,1

Comments

A projectile is launched with an initial speed v at angle theta above the horizon. Assuming that the gravitational acceleration g is uniform and neglecting the air resistance, the trajectory is a part of a parabola whose expected length, averaged over theta uniformly chosen at random from the range [0, Pi/2], is c * v^2/g, where c is this constant.

The length of the trajectory as a function of theta is L(theta) = (v^2/g)*(sin(theta) + cos(theta)^2*log((1+sin(theta))/(1-sin(theta))/2. L(theta) goes from 0 to 1 between theta = 0 and Pi/2. It has a maximum at theta = 0.985514... (A345737), and a unique value at 0 <= theta < 0.599677... (A345739). The average length (c * v^2/g) occurs at theta = 0.5152731296... (29.522975... degrees).

Links

Table of n, a(n) for n=0..86.

Péter Kórus, Notes on Projectile Motion, The American Mathematical Monthly, Vol. 126, No. 4 (2019), pp. 358-360.

Formula

Equals (2 * A006752 + 1)/A000796.

Equals 2 * A143233 + 1.

Example

0.90143169424542823181453643968181856179705159945258...

Mathematica

RealDigits[(2*Catalan + 1)/Pi, 10, 100][[1]]

Crossrefs

Cf. A000796, A006752, A143233, A345737, A345739.

Sequence in context: A110909 A361060 A364830 * A197070 A197333 A226120

Adjacent sequences: A345735 A345736 A345737 * A345739 A345740 A345741

Keyword

nonn,cons

Author

Amiram Eldar, Jun 25 2021

Status

approved