A243908 Johannes Kepler's polyhedron inscribing constant.

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

Offset

-1,1

Comments

Decimal expansion of (5 + 2*sqrt(5))/135.

The finite solid analogy to the plane polygon inscribing constant (A085365).

The five Platonic solids are the tetrahedron, the hexahedron (or cube), the octahedron, the dodecahedron and the icosahedron.

The geometric interpretation is as follows. Begin with a unit sphere. Inscribe a tetrahedron and then inscribe a sphere. Inscribe a cube and then inscribe a sphere. Inscribe an octahedron and then inscribe a sphere. Inscribe a dodecahedron and then inscribe a sphere. Inscribe an icosahedron and then inscribe a sphere. This constant is the radius of this last sphere. Actually, the order in which the five solids are inscribed has no effect on the resulting constant.

Links

Chai Wah Wu, Table of n, a(n) for n = -1..10001

Domingo H. A. and Omar E. Pol, Circunferencias concéntricas y polígonos regulares inscritos, gaussianos, Nov 16 2007, 18:46, 23:28

Wikipedia, Platonic Solids

Wikipedia, Johannes Kepler

Wikipedia, Kepler's solar system

Formula

Equals 1/A211174 = 1/(9*(15 - 6*sqrt(5))).

Example

0.070163970037033921428284054351574462747268420142392973692902181413489198898...

Maple

Digits:=100: evalf((5+2*sqrt(5))/135); # Wesley Ivan Hurt, Sep 05 2014

Mathematica

RealDigits[(5 + 2 Sqrt[5])/135, 10, 100][[1]] (* Wesley Ivan Hurt, Sep 05 2014 *)

Prog

(PARI) first(n)=digits(floor(10^(n+1)*(5+2*sqrt(5))/135)) \\ Edward Jiang, Sep 05 2014

Crossrefs

Cf. A051762, A085365, A211174.

Sequence in context: A169681 A202354 A324498 * A186642 A153626 A297787

Adjacent sequences: A243905 A243906 A243907 * A243909 A243910 A243911

Keyword

nonn,cons

Author

Omar E. Pol, Jun 14 2014

Status

approved