A237603 - OEIS

%I #17 Jan 27 2025 21:14:23

%S 1,1,1,3,5,1,6,3,6,4,4,1,1,6,0,6,7,3,5,1,9,4,3,7,5,0,3,9,4,8,6,9,4,9,

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

%U 8,9,6,2,0,5,6,5,5,6,5,9,7,5,8,8

%N Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

%C Equals phi^2/(2*xi), where phi is the golden ratio (A001622, 2*cos(Pi/5)) and xi is its associate (A182007, 2*sin(Pi/5)).

%D Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

%H Stanislav Sykora, <a href="/A237603/b237603.txt">Table of n, a(n) for n = 1..2000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Platonic_solid">Platonic solid</a>.

%F Equals A001622^2/A182007 = (cos(Pi/5))^2/sin(Pi/5) = A019863^2/A019845 = cos(Pi/5)*cotan(Pi/5) = A019863*A019952 = 1/sin(Pi/5) - sin(Pi/5) = A019845^(-1) - A019845 = sqrt(250+110*sqrt(5))/20.

%e 1.1135163644116067351943750394869493758831503698864877726012080...

%t RealDigits[ Cos[Pi/5]^2 / Sin[Pi/5], 10, 111][[1]] (* Or *)

%t RealDigits[ Sqrt[5/8 + 11/(8 Sqrt[5])], 10, 111][[1]] (* _Robert G. Wilson v_, Feb 28 2014 *)

%o (PARI) sqrt(250+110*sqrt(5))/20

%Y Cf. A001622, A182007, A019863, A019863, A019952, A374771 (sphere volume).

%Y Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A179294 (icosahedron).

%K nonn,cons,easy,changed

%O 1,4

%A _Stanislav Sykora_, Feb 25 2014