A378824 Decimal expansion of the volume of a pentagonal icositetrahedron with unit shorter edge length.

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

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

			35.63020201207128322396774163519636903538669152186...
		

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. A378823 (surface area), A378825 (inradius), A378826 (midradius), A378827 (dihedral angle).

Cf. A377603 (volume of a snub cube with unit edge length).

Cf. A058265.

Programs

Mathematica

First[RealDigits[Root[#^6 - 1269*#^4 - 649*#^2 - 121 &, 2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentagonalIcositetrahedron", "Volume"], 10, 100]]

Formula

Equals 4*(1 + s)^3*(2 + 3*s)*sqrt(1 - 2*s)/((1 + s)*(1 - 4*s^2)), where s = (A058265 - 1)/2.

Equals the positive real root of x^6 - 1269*x^4 - 649*x^2 - 121.