A156546 -id:A156546 - cp's OEIS frontend

A387150 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square pyramid (Johnson solid J_10).

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

Offset: 1

Paolo Xausa, Aug 19 2025

This is the dihedral angle between the square face and a triangular face.

			1.81228288299223868132256212312198395270891707198...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated square pyramid.

Cf. other J_10 dihedral angles: A156546, A387148, A387149.
Cf. A179638 (J_10 volume), A374948 (J_10 surface area).
Cf. A010466.

First[RealDigits[ArcCos[-Sqrt[1 - Sqrt[8]/3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J10", "DihedralAngles"]], 10, 100]]

Equals arccos(-sqrt(1 - 2*sqrt(2)/3)) = arccos(-sqrt(1 - A010466/3)).

A087125 Indices k of hex numbers H(k) that are also triangular.

Original entry on oeis.org

0, 5, 54, 539, 5340, 52865, 523314, 5180279, 51279480, 507614525, 5024865774, 49741043219, 492385566420, 4874114620985, 48248760643434, 477613491813359, 4727886157490160, 46801248083088245, 463284594673392294, 4586044698650834699, 45397162391834954700

Offset: 0

Eric W. Weisstein, Aug 14 2003

From the law of cosines, the non-Pythagorean triple {a(n), a(n)+1=A253475(n+1), A072256(n+1)} forms a near-isosceles triangle with the angle bounded by the consecutive sides equal to the regular tetrahedron's central angle (see A156546 and A247412). This implies also that a(n) are those numbers k such that (16/3)*A000217(k)+1 is a perfect square. - Federico Provvedi, Apr 04 2023

Colin Barker, Table of n, a(n) for n = 0..1000
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
Eric Weisstein's World of Mathematics, Hex Number
Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Cf. A006244, A031138, A072256, A054318.

[Round((-4-(5-2*Sqrt(6))^n*(-2+Sqrt(6)) + (2+Sqrt(6))*(5 + 2*Sqrt(6))^n)/8): n in [0..25]]; // G. C. Greubel, Nov 04 2017

CoefficientList[Series[(-x^2+5*x)/((1-x)*(1-10*x+x^2)), {x, 0, 25}], x] (* G. C. Greubel, Nov 04 2017 *)
LinearRecurrence[{11,-11,1},{0,5,54},30] (* Harvey P. Dale, Jun 14 2022 *)
Table[(x Sqrt[z^(2 n + 1) + z^-(2 n + 1) - 2] - 4) / 8 //. {x -> Sqrt[2], y -> Sqrt[3], z -> (5 + 2 x y)}, {n, 0, 100}] // Round (* Federico Provvedi, Apr 16 2023 *)

concat(0, Vec(x*(x-5)/((x-1)*(x^2-10*x+1)) + O(x^50))) \\ Colin Barker, Jun 23 2015

G.f.: (-x^2+5*x)/((1-x)*(1-10*x+x^2)).
a(n) = 11*a(n-1) - 11*a(n-2) + a(n-3) for n > 2. - Colin Barker, Jun 23 2015
a(n) = (-4 - (5-2*sqrt(6))^n*(-2 + sqrt(6)) + (2+sqrt(6))*(5+2*sqrt(6))^n)/8. - Colin Barker, Mar 05 2016
a(n) = 10*a(n-1) - a(n-2) + 4 for n > 1. - Charlie Marion, Feb 14 2023
a(n) = ((x^(n+1)+1)*(x^n-1))/(2*x^n*(x-1)), with x=5+2*sqrt(6). - Federico Provvedi, Apr 04 2023
a(n) = sqrt(3*A161680(A054318(n+1)) + 1/4) - 1/2 = floor(sqrt(3*A000217(A054318(n+1)-1) + 1/4)). - Federico Provvedi, Apr 16 2023

A378389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a tetrakis hexahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 27 2024

The tetrakis hexahedron is the dual polyhedron of the truncated octahedron.

			2.498091544796508851659834154562180246155658808...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Tetrakis hexahedron.

Cf. A378388 (surface area), A374359 (volume - 1), A010532 (inradius*10), A179587 (midradius + 1).

Cf. A156546 and A195698 (dihedral angles of a truncated octahedron), A195729.

First[RealDigits[ArcCos[-4/5], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TetrakisHexahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-4/5).

Equals 2*A195729. - Amiram Eldar, Nov 27 2024

A289502 Mean width of the regular tetrahedron of unit length.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 07 2017

Steven R. Finch, Mean width of a regular simplex, arxiv:1111.4976 [math.MG] (2016), E(w_3).

Cf. A156546, A093582.

Maple
```
x := 3*arccos(-1/3)/2/Pi ; evalf(%) ;
```

RealDigits[3 ArcCos[-1/3]/(2*Pi), 10, 87][[1]] (* Indranil Ghosh, Jul 08 2017 *)

Equals A156546*A093582.

A336199 Decimal expansion of the distance between the centers of two unit-radius spheres such that the volume of intersection is equal to the sum of volumes of the two nonoverlapping parts.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 11 2020

Solution to the three-dimensional version of Mrs. Miniver's problem.

The intersection volume is equal to 2/3 of the volume of each sphere, i.e., 8*Pi/9.

			0.452147427578415981828610831183181263247509153259677...

Eric Weisstein's World of Mathematics, Sphere-Sphere Intersection.
Wikipedia, Mrs. Miniver's problem.

Cf. A019673, A019699, A133749, A156546, A255899.

RealDigits[4 * Sin[ArcCos[-1/3]/3 - Pi/6], 10, 100][[1]]

Equals 4 * sin(arccos(-1/3)/3 - Pi/6).

The smaller of the two positive roots of the equation x^3 - 12*x + 16/3 = 0.

cp's OEIS Frontend

A387150 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square pyramid (Johnson solid J_10).

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A087125 Indices k of hex numbers H(k) that are also triangular.

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A378389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a tetrakis hexahedron.

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A289502 Mean width of the regular tetrahedron of unit length.

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A336199 Decimal expansion of the distance between the centers of two unit-radius spheres such that the volume of intersection is equal to the sum of volumes of the two nonoverlapping parts.

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