A395847 - OEIS

%I #4 May 09 2026 05:02:19

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

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

%U 3,0,4,2,2,6,1,8,7,2,6,0,8,4,0,6,9,3,7,2,3,4,7,4,9,8,9,3,4,6,6,2,4,6,7,3,2

%N Decimal expansion of the expected distance between the center of a unit-side regular pentagon to a randomly and uniformly selected point on its perimeter.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals (sqrt(10*(5 + sqrt(5))) + (5 + 2*sqrt(5))*arccosh(sqrt(5) - 1))/20.

%F Equals (5+2*sqrt(5))*(tan(Pi/5)*sec(Pi/5) + arcsinh(tan(Pi/5)))/20.

%F Equals (sqrt(5)*phi^3/20)*(2*tan(Pi/5)/phi + log(2/phi + tan(Pi/5))), where phi is the golden ratio (A001622).

%e 0.744666853936194163107459534588181561974365758435389...

%t RealDigits[(Sqrt[10*(5 + Sqrt[5])] + (5 + 2*Sqrt[5])*ArcCosh[Sqrt[5] - 1])/20, 10, 120][[1]]

%o (PARI) (sqrt(10*(5 + sqrt(5))) + (5 + 2*sqrt(5))*acosh(sqrt(5) - 1))/20

%Y Cf. A001622, A394596, A395846 (equilateral triangle), A345653 (square), A395848 (regular hexagon).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, May 08 2026