A020761 Decimal expansion of 1/2.

5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

Real part of all nontrivial zeros of the Riemann zeta function (assuming the Riemann hypothesis to be true). - Alonso del Arte, Jul 02 2011

Radius of a sphere with surface area Pi. - Omar E. Pol, Aug 09 2012

Radius of the midsphere (tangent to the edges) in a regular octahedron with unit edges. Also radius of the inscribed sphere (tangent to faces) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014

Construct a rectangle of maximal area inside an arbitrary triangle. The ratio of the rectangle's area to the triangle's area is 1/2. - Rick L. Shepherd, Jul 30 2014

Examples

			1/2 = 0.50000000000000...
		

Links

• Michael Penn, A creative approach to a scary looking integral., YouTube video, 2020.
• Michael Penn, I really like this sum!, YouTube video, 2021.
• Wikipedia, Riemann zeta function
• Wikipedia, Platonic solid
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. In platonic solids:

midsphere radii:

A020765 (tetrahedron),

A010503 (cube),

A019863 (icosahedron),

A239798 (dodecahedron);

insphere radii:

A020781 (tetrahedron),

A020763 (octahedron),

A179294 (icosahedron),

A237603 (dodecahedron).

Programs

Maple

Digits:=100; evalf(1/2); # Wesley Ivan Hurt, Mar 27 2014

Mathematica

RealDigits[1/2, 10, 128][[1]] (* Alonso del Arte, Dec 13 2013 *)
LinearRecurrence[{1},{5,0},99] (* Ray Chandler, Jul 15 2015 *)

PARI

{ default(realprecision); x=1/2*10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Felix Fröhlich, Jul 24 2014

PARI

a(n) = 5*(n==0); \\ Michel Marcus, Jul 25 2014

Formula

Equals Sum_{k>=1} (1/3^k). Hence 1/2 = 0.1111111111111... in base 3.

Cosine of 60 degrees, i.e., cos(Pi/3).

-zeta(0), zeta being the Riemann function. - Stanislav Sykora, Mar 27 2014

a(0) = 5; a(n) = 0, n > 0. - Wesley Ivan Hurt, Mar 27 2014

a(n) = 5 * floor(1/(n + 1)). - Wesley Ivan Hurt, Mar 27 2014

Equals 2*A019824*A019884. - R. J. Mathar, Jan 17 2021