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

3,2

Comments

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

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.

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.

Example

189.78985206688527910632308619447379699106033629736...

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]]

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.

Sequence in context: A154163 A019601 A337669 * A198222 A154582 A020507

Adjacent sequences: A379886 A379887 A379888 * A379890 A379891 A379892

Keyword

nonn,cons,easy

Author

Paolo Xausa, Jan 07 2025

Status

approved