A010469 -id:A010469 - cp's OEIS frontend

A176394 Decimal expansion of 3+2*sqrt(3).

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

Offset: 1

Klaus Brockhaus, Apr 16 2010

Continued fraction expansion of 3+2*sqrt(3) is A010696 preceded by 6.
a(n) = A010469(n) for n > 1.
Largest radius of three circles tangent to a circle of radius 1. - Charles R Greathouse IV, Jan 14 2013
For a spinning black hole the phase transition to positive specific heat happens at a point governed by 2*sqrt(3)-3 (according to a discussion on John Baez's blog), and not at the golden ratio as claimed by Paul Davis. - Peter Luschny, Mar 02 2013
In particular: a black hole with J > (2*sqrt(3)-3) Gm^2/c has positive specific heat, and negative specific heat if J is less, where J is its angular momentum, m is its mass, G is the gravitational constant, and c is the speed of light. For a solar mass black hole, this is 4.08 * 10^41 joule-seconds or a rotation every 1.61 days with the sun's inertia. - Charles R Greathouse IV, Sep 20 2013

			3+2*sqrt(3) = 6.46410161513775458705...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
John Baez, Black Holes and The Golden Ratio, Mar 01 2013.
Index entries for algebraic numbers, degree 2.

Cf. A002194 (decimal expansion of sqrt(3)), A010469 (decimal expansion of sqrt(12)), A010696 (repeat 2, 6).

Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}], {Blue, Circle[{0, 0}, 1]}]]]; Circs[3] (* Charles R Greathouse IV, Jan 14 2013 *)

3+2*sqrt(3) \\ Charles R Greathouse IV, Jan 14 2013

Equals Sum_{n>=1} (sqrt(3)/2)^n = (sqrt(3)/2)/(1 - (sqrt(3)/2)). - Fred Daniel Kline, Mar 03 2014

A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

			0.7733429420779898501961016...

Muniru A Asiru, Table of n, a(n) for n = 0..2000
R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (3.8).
Wikipedia, Fresnel Integral.

Cf. A010469, A019670, A073005, A073006, A204068, A217481.

evalf(int(cos(x^3),x=0..infinity),120); # Muniru A Asiru, Sep 26 2018

RealDigits[Gamma[1/3]/(2*Sqrt[3]), 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Pi/(3*gamma(2/3)) \\ Gheorghe Coserea, Sep 26 2018

intnum(x=[0, -2/3], [oo, I], cos(x)/x^(2/3))/3 \\ Gheorghe Coserea, Sep 26 2018

Equals Pi/(3*Gamma(2/3)) = A019670 / A073006.

Equals Gamma(1/3)/(2*sqrt(3)) = A073005 / A010469. - Amiram Eldar, May 26 2023

A343964 Decimal expansion of 18 + 2*sqrt(3).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 05 2021

Surface area of a rhombicuboctahedron with unit edge length.

Essentially the same sequence of digits as A176394 and A010469. - R. J. Mathar, May 07 2021

			21.464101615137754587054892683011744733885...

Wikipedia, Rhombicuboctahedron

Cf. A343965 (rhombicuboctahedron volume).

SetDefaultRealField(RealField(100)); 18+2*Sqrt(3);

RealDigits[N[18 + 2*Sqrt[3], 100]][[1]] (* Wesley Ivan Hurt, Nov 12 2022 *)

A387296 Decimal expansion of the third largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 26 2025

This is the dihedral angle between triangular faces in the antiprism part of the solid.

Also the analogous dihedral angle in a gyroelongated triangular bicupola (Johnson solid J_44).

			2.5346001497151261930915028102102107021498303291935...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular bicupola.
Wikipedia, Gyroelongated triangular cupola.

Cf. other J_22 dihedral angles: A195698, A387294, A387295, A387297.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A385256 (J_44 volume), A385257 (J_44 surface area).
Cf. A010469.

First[RealDigits[ArcCos[(1 - Sqrt[12])/3], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J22", "DihedralAngles"]], 3], 10, 100]]

Equals arccos((1 - 2*sqrt(3))/3) = arccos((1 - A010469)/3).

A387297 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 26 2025

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

			1.7261206622946734694269434030970502773414686910539...

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

Cf. other J_22 dihedral angles: A195698, A387294, A387295, A387296.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A010469.

First[RealDigits[ArcCos[1 - 2/Sqrt[3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J22", "DihedralAngles"]], 10, 100]]

Equals arccos(1 - 2*sqrt(3)/3) = arccos(1 - A010469/3).

A194390 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(12) and < > denotes fractional part.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 28, 30, 32, 34, 36, 38, 40, 56, 58, 60, 62, 64, 66, 68, 84, 86, 88, 90, 92, 94, 96, 112, 114, 116, 118, 120, 122, 124, 140, 142, 144, 146, 148, 150, 152, 168, 170, 172, 174, 176, 178, 180

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

Every term is even; see A194368.

Cf. A010469, A194368, A194391.

r = Sqrt[12]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t1, 1]]  (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]       (* A194390 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]       (* A194391 *)

A194391 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(12) and < > denotes fractional part.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 33, 35, 37, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 87, 89, 91, 93

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010469, A194368, A194390.

r = Sqrt[12]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t1, 1]]  (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]       (* A194390 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]       (* A194391 *)

A343199 Decimal expansion of 6+2*sqrt(3).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 07 2021

Decimal expansion of the surface area of a cuboctahedron with unit edge length.

Essentially the same sequence of digits as A176394 and A010469. - R. J. Mathar, Jun 10 2021

			9.4641016151377545870548926830117447338856...

Wikipedia, Cuboctahedron

Cf. A020775 (cuboctahedron volume).

SetDefaultRealField(RealField(200)); 6+2*Sqrt(3);

RealDigits[N[6 + 2*Sqrt[3], 100]][[1]] (* Wesley Ivan Hurt, Nov 12 2022 *)

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.

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.

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A176394 Decimal expansion of 3+2*sqrt(3).

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A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

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A343964 Decimal expansion of 18 + 2*sqrt(3).

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A387296 Decimal expansion of the third largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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A387297 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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A194390 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(12) and < > denotes fractional part.

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A194391 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(12) and < > denotes fractional part.

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A343199 Decimal expansion of 6+2*sqrt(3).

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