A386692 Decimal expansion of the surface area of a parabidiminished rhombicosidodecahedron with unit edges.

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

Offset: 2

Author

Paolo Xausa, Jul 30 2025

Keywords

• nonn
• cons
• easy

Comments

The parabidiminished rhombicosidodecahedron is Johnson solid J_80.

Also the surface area of a metabidiminished rhombicosidodecahedron and a gyrate bidiminished rhombicosidodecahedron (Johnson solids J_81 and J_82, respectively) with unit edges.

Examples

			56.9233187106881294742601885078353260314642655523...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 2..10000
• Wikipedia, Gyrate bidiminished rhombicosidodecahedron.
• Wikipedia, Metabidiminished rhombicosidodecahedron.
• Wikipedia, Parabidiminished rhombicosidodecahedron.

Crossrefs

Cf. A386691 (volume).

Cf. A002194, A010476, A344149, A386690, A386694.

Programs

Mathematica

First[RealDigits[5/2*(8 + Sqrt[3] + 2*Sqrt[#] + Sqrt[5*#]) & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J80", "SurfaceArea"], 10, 100]]

Formula

Equals (5/2)*(8 + sqrt(3) + 2*sqrt(5 + 2*sqrt(5)) + sqrt(5*(5 + 2*sqrt(5)))) = (5/2)*(8 + A002194 + 2*sqrt(5 + A010476) + sqrt(5*(5 + A010476))).

Equals the largest root of x^8 - 160*x^7 + 9000*x^6 - 184000*x^5 - 828750*x^4 + 79100000*x^3 - 718984375*x^2 - 3800625000*x + 55781640625.