A396613 - OEIS

%I #5 May 31 2026 09:45:35

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

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

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

%N Decimal expansion of the mean distance from the center to the interior points of a regular icosahedron of unit volume.

%H <a href="/index/Ia#icosahedron">Index entries for sequences related to icosahedron</a>.

%F Equals (sqrt((5 + sqrt(5))/2)/4 - (9 + 4*sqrt(5))*Pi/180 + (23 + 9*sqrt(5))*arccosech(phi)/24)/(10*(1 + sqrt(5)/3))^(1/3), where phi is the golden ratio (A001622) (Dominik Beck, personal communication).

%e 0.467285914698282339094884603179286140352476116988902...

%t RealDigits[(Sqrt[(5 + Sqrt[5])/2]/4 - (9 + 4*Sqrt[5])*Pi/180 + (23 + 9*Sqrt[5])*ArcCsch[GoldenRatio]/24)/(10*(1 + Sqrt[5]/3))^(1/3), 10, 120][[1]]

%o (PARI) (sqrt((5 + sqrt(5))/2)/4 - (9 + 4*sqrt(5))*Pi/180 + (23 + 9*sqrt(5))*asinh((sqrt(5)-1)/2)/24)/(10*(1 + sqrt(5)/3))^(1/3)

%Y Cf. A135691 (cube), A396610 (tetrahedron), A396611 (octahedron), A396612 (dodecahedron), this constant (icosahedron).

%Y Cf. A001622, A394104.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, May 31 2026