A377299 -id:A377299 - cp's OEIS frontend

A378352 Decimal expansion of the volume of a (small) triakis octahedron with unit shorter edge length.

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

Offset: 1

Paolo Xausa, Nov 23 2024

The (small) triakis octahedron is the dual polyhedron of the truncated cube.

			2.9142135623730950488016887242096980785696718753769...

Eric Weisstein's World of Mathematics, Small Triakis Octahedron.
Wikipedia, Triakis octahedron.

Cf. A378351 (surface area), A378353 (inradius), A201488 (midradius), A378354 (dihedral angle).
Cf. A377299 (volume of a truncated cube with unit edge).
Cf. A156035.
Essentially the same as A002193 and A188582.

First[RealDigits[Sqrt[2] + 3/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisOctahedron", "Volume"], 10, 100]]

Equals sqrt(2) + 3/2 = A002193 + 3/2.

Equals A156035/2. - Hugo Pfoertner, Nov 24 2024

A377298 Decimal expansion of the surface area of a truncated cube with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			32.4346643636148951726751573735281216767216730121...

Eric Weisstein's World of Mathematics, Truncated Cube.
Wikipedia, Truncated cube.

Cf. A377299 (volume), A294968 (circumradius), A010503 (midradius - 1), A377296 (Dehn invariant, negated).

Cf. A002193, A002194, A010469, A010524.

First[RealDigits[2*(6 + Sqrt[72] + Sqrt[3]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCube", "SurfaceArea"], 10, 100]]

Equals 2*(6 + 6*sqrt(2) + sqrt(3)) = 2*(6 + 2*A002193 + A002194) = 12 + 2*A010524 + A010469.

A386459 Decimal expansion of the volume of an augmented truncated cube with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 22 2025

The augmented truncated cube is Johnson solid J_66.

			15.5424723326565069269423398624517230857049...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Augmented truncated cube.

Cf. A386460 (surface area).

Cf. A131594, A179587, A377299, A386411.

First[RealDigits[8 + 16*Sqrt[2]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J66", "Volume"], 10, 100]]

Equals 8 + 16*sqrt(2)/3 = 8 + 16*A131594.
Equals A377299 + A179587.
Equals the largest root of 9*x^2 - 144*x + 64.

A381686 Decimal expansion of the isoperimetric quotient of a truncated cube.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 04 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.61302821107928032110240558144714079708976169223933...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A377298 (surface area), A377299 (volume).

Cf. A000796, A002194, A010524, A156164, A381684.

First[RealDigits[49*Pi*(17 + 12*Sqrt[2])/(2*(6 + 6*Sqrt[2] + Sqrt[3])^3), 10, 100]]

Equals 36*Pi*A377299^2/(A377298^3).

Equals 49*Pi*(17 + 12*sqrt(2))/(2*(6 + 6*sqrt(2) + sqrt(3))^3) = 49*A000796*A156164/(2*(6 + A010524 + A002194)^3).

cp's OEIS Frontend

A378352 Decimal expansion of the volume of a (small) triakis octahedron with unit shorter edge length.

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A377298 Decimal expansion of the surface area of a truncated cube with unit edge length.

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A386459 Decimal expansion of the volume of an augmented truncated cube with unit edges.

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A381686 Decimal expansion of the isoperimetric quotient of a truncated cube.

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