A131594 -id:A131594 - cp's OEIS frontend

A385578 Decimal expansion of the volume of a parabiaugmented hexagonal prism with unit edge.

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

Offset: 1

Paolo Xausa, Jul 04 2025

The parabiaugmented hexagonal prism is Johnson solid J_55.

Also the volume of a metabiaugmented hexagonal prism (Johnson solid J_56) with unit edge.

			3.0694807321443476232250657536620412432707651725...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Parabiaugmented hexagonal prism.

Cf. A385257 (surface area + 2).

Cf. A002194, A010466, A104956, A131594, A385569.

First[RealDigits[(Sqrt[8] + 9*Sqrt[3])/6, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J55", "Volume"], 10, 100]]

Equals (2*sqrt(2) + 9*sqrt(3))/6 = (A010466 + 9*A002194)/6 = A131594 + A104956.

Equals the largest root of 1296*x^4 - 18072*x^2 + 55225.

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.

A386854 Decimal expansion of the largest dihedral angle, in radians, in an elongated triangular pyramid (Johnson solid J_7).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 06 2025

Also the largest dihedral angle, in radians, in an elongated triangular bipyramid, elongated pentagonal bipyramid, elongated triangular orthobicupola and elongated triangular gyrobicupola (Johnson solids J_14, J_18, J_35 and J_36, respectively).

			2.80175574413567130136625086988773881780892470904...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Elongated triangular bipyramid.
Wikipedia, Elongated triangular cupola.
Wikipedia, Elongated triangular gyrobicupola.
Wikipedia, Elongated triangular orthobicupola.
Wikipedia, Elongated triangular pyramid.
Index entries for transcendental numbers.

Cf. A010466, A131594.

First[RealDigits[ArcCos[-Sqrt[8]/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J7", "DihedralAngles"]], 10, 100]]

acos(-sqrt(8)/3) \\ Charles R Greathouse IV, Aug 19 2025

Equals arccos(-sqrt(8)/3) = arccos(-A010466/3) = arccos(-2*A131594).

A353049 Decimal expansion of 8*sqrt(2) / 3.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Apr 20 2022

8*sqrt(2) / (3*a) is the maximum curvature of the Folium of Descartes x^3 + y^3 - 3*a*x*y = 0, occurring at the point M of coordinates (3a/2, 3a/2). The corresponding minimum radius of curvature is (3*sqrt(2))*a/16.

This point M is at the intersection of the first bisector with the loop, distinct from O (see curves).

			3.771236166328253463471169931225...

Robert Ferréol, Cartesian folium, Mathcurve.
John A. Tierney, Problem 417, Crux Mathematicorum, Vol. 5, No. 10 (1979), pp. 308-310.
Eric Weisstein's World of Mathematics, Folium of Descartes.
Wikipedia, Folium of Descartes.
Index to sequences related to curves.

Cf. A295709 (arc length of the loop of the Folium of Descartes).

Cf. A002193, A131594.

Maple
```
evalf(8*sqrt(2)/3,100);
```

RealDigits[8*Sqrt[2]/3, 10, 100][[1]] (* Amiram Eldar, Apr 20 2022 *)

8*sqrt(2)/3 \\ Michel Marcus, Apr 20 2022

Equals 8*A131594.

cp's OEIS Frontend

A385578 Decimal expansion of the volume of a parabiaugmented hexagonal prism with unit edge.

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

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A386854 Decimal expansion of the largest dihedral angle, in radians, in an elongated triangular pyramid (Johnson solid J_7).

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A353049 Decimal expansion of 8*sqrt(2) / 3.

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