A372273 -id:A372273 - cp's OEIS frontend

A382103 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.

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

Offset: 0

A.H.M. Smeets, Mar 15 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                     | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------+----------------------------------------------------
   2 | A020760                   | A000007*10
   3 | A010513/10                | A010716
   4 | A372267, A372268          | this sequence, A382104
   5 | A372269, A372270          | A382105, A382106
   6 | A372271, A372272, A372273 | A382107, A382686, A382687
   7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.34785484513745385737306394922199940723534869583389...

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=4.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372267.

RealDigits[1/2 - Sqrt[5/6]/6, 10, 120][[1]] (* Amiram Eldar, Mar 24 2025 *)

Equals 1/2 - (1/6)*sqrt(5/6).

A382104 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372268.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Mar 15 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                     | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------+----------------------------------------------------
   2 | A020760                   | A000007*10
   3 | A010513/10                | A010716
   4 | A372267, A372268          | A382103, this sequence
   5 | A372269, A372270          | A382105, A382106
   6 | A372271, A372272, A372273 | A382107, A382686, A382687
   7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.65214515486254614262693605077800059276465130416610645...

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=4.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372268.

RealDigits[1/2 + Sqrt[5/6]/6, 10, 120][[1]] (* Amiram Eldar, Mar 24 2025 *)

1/2 + (1/6)*sqrt(5/6) \\ Stefano Spezia, May 22 2025

Equals 1/2 + (1/6)*sqrt(5/6).

Minimal polynomial: 216*x^2 - 216*x + 49. - Stefano Spezia, May 22 2025

A382106 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.

Original entry on oeis.org

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

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          | A382106, this sequence
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.236926885056189087514264040719917362643260002212...

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=5.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372270.

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

A382107 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.

Original entry on oeis.org

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

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          | A382105, A382106
 6 | A372271, A372272, A372273 | this sequence, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.4679139345726910473898703439895509948116556057692...

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. A372271.

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

Original entry on oeis.org

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.

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.

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.

cp's OEIS Frontend

A382103 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.

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A382104 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372268.

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A382106 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.

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A382107 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.

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

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

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