A247412 -id:A247412 - cp's OEIS frontend

A156546 Decimal expansion of the central angle of a regular tetrahedron.

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

Offset: 1

Clark Kimberling, Feb 09 2009

If O is the center of a regular tetrahedron ABCD, then the central angle AOB is this number; exact value is Pi - arccos(1/3).
The (minimal) central angle of the other four regular polyhedra are as follows:
- cube: A137914,
- octahedron: A019669,
- dodecahedron: A156547,
- icosahedron: A105199.
Dihedral angle of two adjacent faces of the octahedron. - R. J. Mathar, Mar 24 2012
Best known as "tetrahedral angle" theta (e.g., in chemistry). Its Pi complement (i.e., Pi - theta) is the dihedral angle between adjacent faces in regular tetrahedron. - Stanislav Sykora, May 31 2012
Also twice the magic angle (A195696). - Stanislav Sykora, Nov 14 2013

			Pi - arccos(1/3) = 1.910633236249018556..., or, in degrees, 109.471220634490691369245999339962435963006843100... = A247412

Wikipedia, Tetrahedron
Wikipedia, Tetrahedral molecular geometry
Index entries for transcendental numbers.

Cf. A195696, A247412.

Cf. Platonic solids dihedral angles: A137914 (tetrahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron). - Stanislav Sykora, Jan 23 2014

evalf(arccos(-1/3)) ; # R. J. Mathar, Feb 18 2025

RealDigits[Pi-ArcCos[1/3],10,120][[1]] (* Harvey P. Dale, Aug 25 2011 *)

acos(-1/3) \\ Charles R Greathouse IV, Aug 30 2013

Start with vertices (1,1,1), (1,-1,-1,), (-1,1,-1), and (1,-1,1) and apply the formula for cosine of the angle between two vectors.
Equals 2* A195696. - R. J. Mathar, Mar 24 2012
Equals A000796 - A137914 = A247412 / A072097 - R. J. Mathar, Feb 18 2025

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

cp's OEIS Frontend

A156546 Decimal expansion of the central angle of a regular tetrahedron.

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