A377342 -id:A377342 - cp's OEIS frontend

A378354 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a (small) triakis octahedron.

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

Offset: 1

Paolo Xausa, Nov 24 2024

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

			2.57174440034566846791285405092806379355115694111...

Wikipedia, Triakis octahedron.

Cf. A378351 (surface area), A378352 (volume), A378353 (inradius), A201488 (midradius).
Cf. A019669 and A195698 (dihedral angles of a truncated cube).
Cf. A377342.

First[RealDigits[ArcCos[-(3 + 8*Sqrt[2])/17], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TriakisOctahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-(3 + 8*sqrt(2))/17) = arccos(-(3 + A377342)/17).

A377341 Decimal expansion of the surface area of a truncated octahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			26.78460969082652752232935609807046840331366304572...

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

Cf. A377342 (volume), A020797 (circumradius/10), A152623 (midradius).
Cf. A010469 (analogous for a regular octahedron).
Cf. A002194.

First[RealDigits[6 + 12*Sqrt[3], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedOctahedron", "SurfaceArea"], 10, 100]]

Equals 6 + 12*sqrt(3) = 6 + 12*A002194.

A386461 Decimal expansion of the surface area of a biaugmented truncated cube with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 23 2025

The biaugmented truncated cube is Johnson solid J_67.

			36.241911729260269564523295159701074096328596018257...

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

Cf. A010524 (volume - 9).

Cf. A010469, A010487, A010502, A377342, A385535, A386460.

First[RealDigits[18 + 8*Sqrt[2] + Sqrt[48], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J67", "SurfaceArea"], 10, 100]]

Equals 2*(9 + 4*sqrt(2) + 2*sqrt(3)) = 2*(9 + A010487 + A010469) = 18 + A377342 + A010502.

Equals the largest root of x^4 - 72*x^3 + 1592*x^2 - 10656*x - 2672.

A381687 Decimal expansion of the isoperimetric quotient of a truncated octahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 04 2025

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

			0.753366625166156882229489414578751361927704595866...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377341 (surface area), A377342 (volume).

Cf. A000796, A010469, A381684.

First[RealDigits[64*Pi/(3*(1 + 2*Sqrt[3])^3), 10, 100]]

Equals 36*Pi*A377342^2/(A377341^3).

Equals 64*Pi/(3*(1 + 2*sqrt(3))^3) = 64*A000796/(3*(1 + A010469)^3).

A379214 Decimal expansion of (sqrt(3) + sqrt(5) + 2sqrt(6))/(8sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 23 2024

			0.78374816457669166276912268657263121209104522720146...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.6, p. 505.

Cf. A002163, A002194, A010464, A010480, A377342.

RealDigits[(Sqrt[3]+Sqrt[5]+2Sqrt[6])/(8Sqrt[2]),10,100][[1]]

Minimal polynomial: 16777216*x^8 - 16777216*x^6 + 4595712*x^4 - 385024*x^2 + 2401. - Stefano Spezia, Aug 03 2025

cp's OEIS Frontend

A378354 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a (small) triakis octahedron.

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A377341 Decimal expansion of the surface area of a truncated octahedron with unit edge length.

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A386461 Decimal expansion of the surface area of a biaugmented truncated cube with unit edges.

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A381687 Decimal expansion of the isoperimetric quotient of a truncated octahedron.

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A379214 Decimal expansion of (sqrt(3) + sqrt(5) + 2*sqrt(6))/(8*sqrt(2)).

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A379214 Decimal expansion of (sqrt(3) + sqrt(5) + 2sqrt(6))/(8sqrt(2)).