A020760 -id:A020760 - cp's OEIS frontend

A382686 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, this sequence, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.36076157304813860756983351383771611166152189274674...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=6.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372272.

A382687 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372273.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, this sequence
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.17132449237917034504029614217273289352682250148404...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=6.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372273.

A382688 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372274.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | this sequence, A382689, A382690

			0.3818300505051189449503697754889751338783650835338627...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372274.

A382689 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372275.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, this sequence, A382690

			0.279705391489276667901467771423779582486925065226598764...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372275.

A382690 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372276.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, this sequence

			0.12948496616886969327061143267908201832858740225994666...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372276.

A020784 Decimal expansion of 1/sqrt(27).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This is the minimum ripple factor for a third-order Chebyshev filter for which the generalized reflectionless topology needs no negative elements. - Matthew A. Morgan, Oct 18 2017

			0.1924500897298752548363829268339858185492005837567089586728674....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.16, pp. 495, 527.

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

Cf. A002194, A010482, A020760, A021031, A073010, A212886.

1/Sqrt(27); // G. C. Greubel, Feb 15 2018

RealDigits[1/Sqrt[27],10,200][[1]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)
realDigitsRecip[Sqrt[27]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jul 05 2025 *)

1/sqrt(27) \\ G. C. Greubel, Feb 15 2018

Equals Sum_{k>=0} binomial(2*k,k) * k/16^k. - Amiram Eldar, Aug 02 2020
Equals sqrt(3)/9. - Stefano Spezia, Dec 24 2024
Equals 1/A010482 = A020760/3 = sqrt(A021031) = A073010/Pi = A212886/2. - Hugo Pfoertner, Dec 24 2024

A382105 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Mar 27 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | this sequence, A382106
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.47862867049936646804129151483563819291229555334...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, (1972) Table 25.4, n=5.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372269.

Equals (322+13*sqrt(70))/900.

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)

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

A097337 Integer part of the edge of a cube that has space-diagonal n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 43

Offset: 1

Cino Hilliard, Sep 17 2004

The first few terms are the same as A038128. However, A038128 is generated by Euler's constant = 0.5772156649015328606065120901..., which is close but not equal to 1/sqrt(3) = 0.5773502691896257645091487805..., which generates this sequence. Euler/(1/sqrt(3)) = 0.9997668585341064519813571911... and the equality fails in the 97th term.

The integers k such that a(k) = a(k+1) give A054406. - Michel Marcus, Nov 01 2021

The Universal Encyclopedia of Mathematics, English translation, 1964, p. 155.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..1000
Index entries for sequences related to Beatty sequences

Cf. A020760 (1/sqrt(3)), A054406.

f(n) = for(x=1,n,s=x\sqrt(3);print1(s","));s

a(n)=sqrtint(n^2\3) \\ Charles R Greathouse IV, Nov 01 2021

Let L be the length of the edges. Then sqrt(2)*L is the diagonal of a face. Whence n^2 = 2*L^2 + L^2, or n = sqrt(3)*L and L = n/sqrt(3).

cp's OEIS Frontend

A382686 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.

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A382687 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372273.

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A382688 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372274.

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A382689 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372275.

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A382690 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372276.

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A020784 Decimal expansion of 1/sqrt(27).

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A382105 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.

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A020763 Decimal expansion of 1/sqrt(6).

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

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