A379889 Decimal expansion of the volume of a pentagonal hexecontahedron with unit shorter edge length.

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

Offset: 3

Author

Paolo Xausa, Jan 07 2025

Keywords

• nonn
• cons
• easy

Comments

The pentagonal hexecontahedron is the dual polyhedron of the snub dodecahedron.

Examples

			189.78985206688527910632308619447379699106033629736...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 3..10000
• Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
• Wikipedia, Pentagonal hexecontahedron.
• Index entries for algebraic numbers, degree 12.

Crossrefs

Cf. A379888 (surface area), A379890 (inradius), A379891 (midradius), A379892 (dihedral angle).

Cf. A377805 (volume of a snub dodecahedron with unit edge length).

Cf. A001622.

Programs

Mathematica

First[RealDigits[Root[3936256*#^12 - 143719449600*#^10 + 69717538560000*#^8 - 965464153000000*#^6 - 5195593956250000*#^4 - 6093827421875000*#^2 + 171855712890625 &, 8], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentagonalHexecontahedron", "Volume"], 10, 100]]

Formula

Equals 5*(1 + t)*(2 + 3*t)/((1 - 2*t^2)*sqrt(1 - 2*t)), where t = ((44 + 12*A001622*(9 + sqrt(81*A001622 - 15)))^(1/3) + (44 + 12*A001622*(9 - sqrt(81*A001622 - 15)))^(1/3) - 4)/12.

Equals the largest real root of 3936256*x^12 - 143719449600*x^10 + 69717538560000*x^8 - 965464153000000*x^6 - 5195593956250000*x^4 - 6093827421875000*x^2 + 171855712890625.