A396613 Decimal expansion of the mean distance from the center to the interior points of a regular icosahedron of unit volume.

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, 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, 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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Index entries for sequences related to icosahedron.

Formula

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).

Example

0.467285914698282339094884603179286140352476116988902...

Mathematica

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]]

Prog

(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)

Crossrefs

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

Cf. A001622, A394104.

Sequence in context: A136323 A135798 A197010 * A021218 A298974 A344983

Adjacent sequences: A396610 A396611 A396612 * A396614 A396615 A396616

Keyword

nonn,cons

Author

Amiram Eldar, May 31 2026

Status

approved