A157697 -id:A157697 - cp's OEIS frontend

A115754 Decimal expansion of sqrt(3/2).

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

Offset: 1

Eric Desbiaux, Jul 30 2008

Coordinate of a control point for a degree-5 integration formula for 7 points over the unit circle. [Stroud & Secrest]
Also real and imaginary part of sqrt(-3i). - Alonso del Arte, Dec 11 2012
Area of the quadrilateral obtained when slicing a unit cube with a plane passing through opposite vertices and the middle of opposite edges. See CNRS link. - Michel Marcus, Mar 26 2016
Positive zero of the Hermite polynomial of degree 3. - A.H.M. Smeets, Jun 02 2025

			1.2247448713915890490986420373529456959829737403283350642163...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Ana Rechtman, Mars 2016, 3e défi, Images des Mathématiques, CNRS, 2016.
A. H. Stroud and D. Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Index entries for algebraic numbers, degree 2

Cf. A382713 (continued fraction), A068388 (Engel expansion).

Cf. A010464 (double), A187110 (half), A157697 (reciprocal).

RealDigits[Sqrt[3/2], 10, 105][[1]] (* Alonso del Arte, Dec 11 2012 *)

Equals 2*A187110.
Equals Sum_{k>=0} binomial(1/2, k)/2^k. - Bruno Berselli, Sep 11 2015
From Amiram Eldar, Aug 02 2020: (Start)
Equals Product_{k>=0} (1 + (-1)^k/(6*k + 3)).
Equals Sum_{k>=0} binomial(2*k,k)/12^k.
Equals 1 + Sum_{k>=1} (2*k - 1)!!/((2*k)!! * 3^k). (End)
Equals A010464/2. - R. J. Mathar, Feb 23 2021

A019693 Decimal expansion of 2*Pi/3.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Volume between a cylinder and the inscribed sphere of radius 1. - Omar E. Pol, Sep 25 2013
(2/3)*Pi is also the surface area of a sphere whose diameter equals the square root of 2/3. More generally, x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Jun 18 2018
Volume of a hemisphere of radius 1. - Omar E. Pol, Aug 17 2019
Angle in radians between two vectors of equal magnitude and originating from the same point whose vector sum also has the same magnitude. - Stefano Spezia, Jun 30 2025

			2.094395102393195492308428922186335256131446... - _Omar E. Pol_, Sep 25 2013

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, pp. 235-236.
Index entries for transcendental numbers.

Cf. A000796, A019692, A019699, A157697.

RealDigits[N[2*Pi/3, 6!]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

PARI
```
2*Pi/3 \\ Michel Marcus, Jun 19 2018
```

Equals 5*A019699. - Omar E. Pol, Aug 17 2019
Equals arccos(-1/2). - Amiram Eldar, Aug 12 2020
Equals Integral_{x=-oo..oo} 1/(1 + x^6) dx. - Stefano Spezia, Mar 05 2022
Equals Integral_{x=0..2*Pi} 1/(5 - 4*cos(x)) dx. - Kritsada Moomuang, May 22 2025

A020763 Decimal expansion of 1/sqrt(6).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Radius of the inscribed sphere (tangent to all faces) in a regular octahedron with unit edge. - Stanislav Sykora, Nov 21 2013

			0.408248290463863016366214012450981898660991246776111688072115427875...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A010464, A157697.

Cf. Platonic solids in radii: A020781 (tetrahedron), A179294 (icosahedron), A237603 (dodecahedron). - Stanislav Sykora, Feb 25 2014

evalf(1/sqrt(6)); # Michal Paulovic, Dec 09 2022

RealDigits[1/Sqrt[6], 10, 100][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
realDigitsRecip[Sqrt[6]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jan 10 2025 *)

From Michal Paulovic, Dec 09 2022: (Start)
Equals A157697/2 = A010503 * A020760 = 1/A010464.
Equals [0, 2; 2, 4] (periodic continued fraction expansion). (End)

A092259 Numbers that are congruent to {4, 8} mod 12.

Original entry on oeis.org

4, 8, 16, 20, 28, 32, 40, 44, 52, 56, 64, 68, 76, 80, 88, 92, 100, 104, 112, 116, 124, 128, 136, 140, 148, 152, 160, 164, 172, 176, 184, 188, 196, 200, 208, 212, 220, 224, 232, 236, 244, 248, 256, 260, 268, 272, 280, 284, 292, 296, 304, 308, 316, 320, 328, 332

Offset: 1

Giovanni Teofilatto, Feb 19 2004

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001651, A017617, A017569, A069587, A092899, A145439, A157697.

Fourth row of A092260.

[n : n in [0..400] | n mod 12 in [4, 8]]; // Wesley Ivan Hurt, May 21 2016

A092259:=n->6*n-3-(-1)^n: seq(A092259(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[6n-3-(-1)^n, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
LinearRecurrence[{1,1,-1},{4,8,16},60] (* Harvey P. Dale, Oct 07 2021 *)

G.f.: 4*x*(1+x+x^2) / ( (1+x)*(x-1)^2 ).
a(n) = 4 * A001651(n).
Iff phi(n) = phi(3n/2), then n is in A069587. - Labos Elemer, Feb 25 2004
a(n) = 12*(n-1)-a(n-1) (with a(1)=4). - Vincenzo Librandi, Nov 16 2010
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = 6n - 3 - (-1)^n.
a(2n) = A017617(n-1) for n>1, a(2n-1) = A017569(n-1) for n>1.
a(n) = -a(1-n), a(n) = A092899(n) + 1 for n>0. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi*sqrt(3)/36. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1/sqrt(2) + 1/sqrt(6) (A145439).
Product_{n>=1} (1 + (-1)^n/a(n)) = sqrt(2/3) (A157697). (End)

Edited and extended by Ray Chandler, Feb 21 2004

A195695 Decimal expansion of arcsin(sqrt(1/3)) and of arccos(sqrt(2/3)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

The complementary magic angle, that is, Pi/2 - A195696. The angle between the body-diagonal and a congruent face-diagonal of a cube. And also the polar angle of the cone circumscribed to a regular tetrahedron from one of its vertices. - Stanislav Sykora, Nov 21 2013

This is the value of the angle of the circular cone to the axis, that maximizes the volume of the cone enclosed by a given area. See the +plus link. - Michel Marcus, Aug 27 2017

			arcsin(sqrt(1/3)) = 0.61547970867038734106746458912399...

G. C. Greubel, Table of n, a(n) for n = 0..5000
John D. Barrow, Outer space: Archimedean ice cream cones, +plus magazine.
Wikipedia, Polyhedron, and further links therein.
Index entries for transcendental numbers

Cf. A195696 (magic angle), A195698, A020760, A157697, A243445.

[Arcsin(Sqrt(1/3))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/3];
N[ArcSin[r], 100]
RealDigits[%]  (* A195695 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195696 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A019673 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195698 *)

atan(1/sqrt(2)) \\ Michel Marcus, Aug 27 2017

Also equals arctan(1/sqrt(2)). - Michel Marcus, Aug 27 2017

A171974 Integer part of the height of a regular tetrahedron with edge length n.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 53, 53, 54, 55, 56, 57, 57, 58, 59

Offset: 1

Reinhard Zumkeller, Jan 20 2010

nonn

-3 <= 4*A171975(n) - 3*a(n) < 3;
a(n)*A171975(n) <= A007590(n);
floor(a(n)*A171971(n)/3) <= A171973(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Tetrahedron
Wikipedia, Tetrahedron
Index entries for sequences related to Beatty sequences

Cf. A171972, A022840. Beatty sequence of A157697.

Cf. A007590, A171971, A171973, A171975.

a171974 = floor . (/ 3) . (* sqrt 6) . fromInteger
-- Reinhard Zumkeller, Dec 15 2012

a(n) = floor(n*sqrt(6)/3).

A239049 Decimal expansion of Pi*(2/3)^(1/2).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 16 2014

Decimal expansion of Pi*6^(1/2)/3.
Constant found in the Hardy-Ramanujan asymptotic formula of the number of partitions of n, for n = 1.
Also constant mentioned in the DeSalvo-Pak paper, see pages 2, 4, 6.

			2.5650996603237281910880727193420128229345213351281846...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 117.

G. C. Greubel, Table of n, a(n) for n = 1..10000
S. DeSalvo and I. Pak, Log-concavity of the partition function, arXiv:1310.7982v1 [math.CO], 2013-2014.
Steven Finch, Integer Partitions, Sep 22 2004. [Cached copy, with permission of the author]
Index entries for transcendental numbers

Cf. A000796.

R:=RealField(); Pi(R)*Sqrt(2/3); // G. C. Greubel, Mar 31 2018

evalf(Pi*(2/3)^(1/2), 120) # Vaclav Kotesovec, Oct 17 2014

RealDigits[Pi*Sqrt[2/3], 10, 100][[1]] (* G. C. Greubel, Mar 31 2018 *)

Pi*sqrt(2/3) \\ G. C. Greubel, Mar 31 2018

Equals A000796 * A157697.

More terms from Vaclav Kotesovec, Oct 17 2014

A219965 Number of non-intersecting unit cubes regularly packed into the tetrahedron of edge length n.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 19, 30, 45, 66, 94, 130, 172, 221, 278, 344, 422, 511, 611, 723, 848, 987, 1140, 1308, 1491, 1691, 1909, 2146, 2401, 2673, 2965, 3278, 3614, 3974, 4355, 4759, 5186, 5638, 6117, 6623, 7156, 7716, 8305, 8923, 9571, 10249, 10958, 11700, 12475, 13285, 14127, 15003, 15914, 16862, 17849, 18874, 19937, 21037, 22177, 23358, 24581, 25846, 27153, 28504

Offset: 1

R. J. Mathar, Dec 02 2012

nonn

The tetrahedron may be aligned with the Cartesian axes by putting its triangular basis on the horizontal plane, with four vertices at (x, y, z) = (0, 0, 0), (n, 0, 0),  (n/2, sqrt(3)*n/2, 0) and (n/2, n/(2*sqrt(3)), n*sqrt(2/3)) see A194082, A020769, A157697.
The volume of tetrahedron is a third times the area of the base triangle times height, (1/3) * (sqrt(3)*n^2/4) * n*sqrt(2/3) = n^3/(3*2^(3/2)) = A020829*n^3. This defines an obvious upper limit of floor(n^3/sqrt(72)) = A171973(n) of placing unit cubes into this tetrahedron.
Regular packing: We place the first layer of unit cubes so they touch the floor of the tetrahedron. Their number is limited by the area of the triangular horizontal section of the plane z=1 inside the tetrahedron, which touches all of them; this isosceles horizontal triangle has edge length E(n,z) = n-z*sqrt(3/2). This edge length is a linear interpolation for triangular horizontal cuts between z=0 at the bottom and the summit of the tetrahedron at z=n*sqrt(2/3).
This first layer confined by a triangle characterized by E(n,z) may host RegSquInTri(E) := sum_{y=1..floor(E*sqrt(3)/2)} floor(E-y*2/sqrt(3)) cubes, following recursively the same regular placement and counting strategy as for squares in isosceles triangles, see A194082.
The number of unit cubes in the next layer, between z=1 and z=2, is limited by the area of the horizontal section of the triangle z=2 inside the tetrahedron, where the triangle has edge length n-z*sqrt(3/2).
So in layer z=1, 2, ... we insert ReqSquInTri(E(n,z)) cubes. a(n) is the sum over all these layers with z limited by the z-value of the vertex at the summit.
There is a generalization to placing unit cubes of higher dimensions into higher dimensional tetrahedra.
The growth is expected to be roughly equal to the growth of A000292.

R. J. Mathar, Illustration of 30 cubes at n=8
R. J. Mathar, Illustration of 66 cubes at n=10
R. J. Mathar, Illustration of 221 cubes at n=14

# Number or squares in isosceles triangle of edge length n.
RegSquInTri := proc(n)
        add(floor(n-2*y/sqrt(3)), y=1..floor(n*sqrt(3)/2)) ;
end proc:
A219965 := proc(n)
        local a,z,triedg ;
        a := 0 ;
        for z from 1 to floor(n*sqrt(2/3)) do
                triedg := n-z*sqrt(3/2) ;
                a := a+ RegSquInTri(triedg) ;
        end do:
        return a;
end proc:

a(n) <= A171973(n).

A368088 Index of smallest pentagonal number with n digits.

Original entry on oeis.org

1, 3, 9, 26, 82, 259, 817, 2583, 8166, 25821, 81650, 258200, 816497, 2581990, 8164966, 25819890, 81649659, 258198890, 816496582, 2581988898, 8164965810, 25819888975, 81649658093, 258198889748, 816496580928, 2581988897472, 8164965809278, 25819888974717, 81649658092773

Offset: 1

Kelvin Voskuijl, Dec 17 2023

The digits of the odd- and even-indexed terms converge to those in the decimal expansions of sqrt(2/3) and sqrt(20/3), respectively.

			a(4) = 26 as the 26th pentagonal number is 26*(3*26-1)/2 = 1001 which has 4 digits (while the 25th is 925 which is only 3).

Cf. A000326, A180447.
Cf. A068092 (for triangular numbers), A017936 (for squares).
Cf. A157697 (square root of 2/3), A020772 (square root of 20/3)

a[n_] := Ceiling[(Sqrt[24*10^(n-1) + 1] + 1)/6]; Array[a, 40] (* Amiram Eldar, Dec 30 2023 *)

a(n) = 1 + (sqrtint(24*10^(n-1)) + 1)\6 \\ Andrew Howroyd, Dec 30 2023

a(n) = ceiling((sqrt(24*10^(n-1) + 1) + 1)/6).

cp's OEIS Frontend

A115754 Decimal expansion of sqrt(3/2).

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A019693 Decimal expansion of 2*Pi/3.

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A020763 Decimal expansion of 1/sqrt(6).

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A092259 Numbers that are congruent to {4, 8} mod 12.

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Magma

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Mathematica

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Extensions

A195695 Decimal expansion of arcsin(sqrt(1/3)) and of arccos(sqrt(2/3)).

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Magma

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PARI

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A171974 Integer part of the height of a regular tetrahedron with edge length n.

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A239049 Decimal expansion of Pi*(2/3)^(1/2).

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