A379888 Decimal expansion of the surface area of a pentagonal hexecontahedron with unit shorter edge length.

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

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

			162.69896419846662676872582412137959709718223664038...
		

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. A379889 (volume), A379890 (inradius), A379891 (midradius), A379892 (dihedral angle).

Cf. A377804 (surface area of a snub dodecahedron with unit edge length).

Cf. A001622.

Programs

Mathematica

First[RealDigits[Root[961*#^12 - 33925050*#^10 + 238487439375*#^8 - 374285139187500*#^6 + 215543322643359375*#^4 - 200764566730722656250*#^2 + 19088214930090087890625 &, 8], 10, 100]]  (* or *)
First[RealDigits[PolyhedronData["PentagonalHexecontahedron", "SurfaceArea"], 10, 100]]

Formula

Equals 30*(2 + 3*t)*sqrt(1 - t^2)/(1 - 2*t^2), 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 961*x^12 - 33925050*x^10 + 238487439375*x^8 - 374285139187500*x^6 + 215543322643359375*x^4 - 200764566730722656250*x^2 + 19088214930090087890625.