A378823 Decimal expansion of the surface area of a pentagonal icositetrahedron with unit shorter edge length.

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

Offset: 2

Author

Paolo Xausa, Dec 09 2024

Keywords

• nonn
• cons
• easy

Comments

The pentagonal icositetrahedron is the dual polyhedron of the snub cube.

Examples

			54.79654943865969339763502315259019090870864439852...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 2..10000
• Eric Weisstein's World of Mathematics, Pentagonal Icositetrahedron.
• Wikipedia, Pentagonal icositetrahedron.

Crossrefs

Cf. A378824 (volume), A378825 (inradius), A378826 (midradius), A378827 (dihedral angle).

Cf. A377602 (surface area of a snub cube with unit edge length).

Cf. A058265.

Programs

Mathematica

First[RealDigits[Root[#^6 - 3060*#^4 + 185328*#^2 - 39517632 &, 2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentagonalIcositetrahedron", "SurfaceArea"], 10, 100]]

Formula

Equals 24*(1 + s)^2*(2 + 3*s)/(1 + 2*s)*sqrt((1 - s)/(1 + s)), where s = (A058265 - 1)/2.

Equals the positive real root of x^6 - 3060*x^4 + 185328*x^2 - 39517632.