A384587 -id:A384587 - cp's OEIS frontend

A384589 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384587.

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

Offset: 0

A.H.M. Smeets, Jun 14 2025

			0.00053929470556132745010379056762059321227725696643324...

Paolo Xausa, 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 25.9, n = 4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

Cf. A384590.
There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279                | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, A384588, this sequence

First[RealDigits[Root[1990656*#^4 - 1990656*#^3 + 504576*#^2 - 16960*# + 9 &, 1], 10, 100, -1]] (* Paolo Xausa, Jun 26 2025 *)

solve(x = 0.0, 0.01, 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9)

Smallest root of 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9 = 0.

A384277 Decimal expansion of the smallest zero of the Laguerre polynomial of degree 3.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, May 24 2025

			0.41577455678347908331153387312827447354661741269311...

Paolo Xausa, 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 25.9, n = 3.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 3.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | this sequence, A384278, A384279          | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, A384588, A384589

First[RealDigits[Root[LaguerreL[3, #] &, 1], 10, 100]] (* Paolo Xausa, Jun 05 2025 *)

Smallest root of x^3 - 9 x^2 + 18 x - 6 = 0.

A384279 Decimal expansion of the largest zero of the Laguerre polynomial of degree 3.

Original entry on oeis.org

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

Offset: 1

A.H.M. Smeets, May 26 2025

			6.28994508293747919686641576551213165749352086624660...

Paolo Xausa, Table of n, a(n) for n = 1..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 25.9, n = 3.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 3.

Cf. A384590.
There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, this sequence          | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, A384588, A384589

First[RealDigits[Root[LaguerreL[3, #] &, 3], 10, 100]] (* Paolo Xausa, Jun 05 2025 *)

largest root of x^3 - 9 x^2 + 18 x - 6 = 0.

A384278 Decimal expansion of the second smallest zero of the Laguerre polynomial of degree 3.

Original entry on oeis.org

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

Offset: 1

A.H.M. Smeets, May 24 2025

			2.29428036027904171982205036135959386895986172106028...

Paolo Xausa, Table of n, a(n) for n = 1..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 25.9, n = 3.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 3.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, this sequence, A384279          | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, A384588, A384589

First[RealDigits[Root[LaguerreL[3, #] &, 2], 10, 100]] (* Paolo Xausa, Jun 05 2025 *)

Second smallest root of x^3 - 9 x^2 + 18 x - 6 = 0.

A384586 Decimal expansion of the second largest zero of the Laguerre polynomial of degree 4.

Original entry on oeis.org

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

Offset: 1

A.H.M. Smeets, Jun 04 2025

			4.53662029692112798327928538495713788012578435338680...

Paolo Xausa, Table of n, a(n) for n = 1..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 25.9, n = 4.
V. I. Krylov, Approximate calculation of integrals (Dover publications) (1962) page 347 n=4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279                | A384463, A384464, A384465
   4 | A384280, A384281, this sequence, A384587 | A384466, A384467, A384588, A384589

First[RealDigits[Root[LaguerreL[4, #] &, 3], 10, 100]] (* Paolo Xausa, Jun 18 2025 *)

solve(x = 2, 6, x^4 - 16*x^3 + 72*x^2 - 96*x + 24)

Second largest root of x^4 - 16 x^3 + 72 x^2 - 96 x + 24 = 0.

A384588 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384586.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Jun 07 2025

			0.038887908515005384272438168156209913722307191348276...

Paolo Xausa, 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 25.9, n = 4.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.
Index entries for algebraic numbers, degree 4.

There are k positive real zeros of the Laguerre polynomial of degree k:
   k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
  ---+------------------------------------------+-----------------------------------------------------
   2 | A101465, 1+A014176                       | A201488, A100954-3
   3 | A384277, A384278, A384279                | A384463, A384464, A384465
   4 | A384280, A384281, A384586, A384587       | A384466, A384467, this sequence, A384589

First[RealDigits[Root[1990656*#^4 - 1990656*#^3 + 504576*#^2 - 16960*# + 9 &, 2], 10, 100, -1]] (* Paolo Xausa, Jun 26 2025 *)

solve(x = 0.1, 0.04, 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9)

Second smallest root of 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9 = 0.

A384590 a(n) = floor(X(n,n)), where X(n,n) is the largest zero of the Laguerre polynomial of degree n.

Original entry on oeis.org

1, 3, 6, 9, 12, 15, 19, 22, 26, 29, 33, 37, 40, 44, 48, 51, 55, 59, 62, 66, 70, 73, 77, 81, 85, 89, 92, 96, 100, 104, 107, 111, 115, 119, 123, 126, 130, 134, 138, 142, 146, 149, 153, 157, 161, 165, 169, 172, 176, 180, 184, 188, 192, 196, 199, 203, 207, 211

Offset: 1

A.H.M. Smeets, Jun 14 2025

nonn

For X(k,n), the k-th smallest zero of the Laguerre polynomial of degree n, see formula section of A091476, for large n and relative small k, k << n.
Some terms for large n:
a(1000) = floor(3943.2473948452...), a(2000) = floor(7927.9014222639...), a(4000) = floor(15908.5812117320...), a(8000) = floor(31884.2511300626...), a(16000) = floor(63853.6067816122...), a(32000) = floor(127815.0051094389...), a(64000) = floor(255766.3763209512...), a(128000) = floor(511705.1129209706...), a(256000) = floor(1023627.9299056501...), a(512000) = floor(2047530.6886230061...).

A.H.M. Smeets, Table of n, a(n) for n = 1..12245
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], Table 25.9.
A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.
Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.

Cf. A091476.

Cf. 1+A014176 (n=2), A384279 (n=3), A384587 (n=4).

A384590[n_] := Floor[Root[LaguerreL[n, #] &, n]];
Array[A384590, 70] (* Paolo Xausa, Jun 26 2025 *)

Limit_{n -> oo} X(n,n)/n = 4.

a(n) ~ floor(4*n + 2 - 5.8917*n^(1/3)).

cp's OEIS Frontend

A384589 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384587.

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A384277 Decimal expansion of the smallest zero of the Laguerre polynomial of degree 3.

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A384279 Decimal expansion of the largest zero of the Laguerre polynomial of degree 3.

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A384278 Decimal expansion of the second smallest zero of the Laguerre polynomial of degree 3.

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A384586 Decimal expansion of the second largest zero of the Laguerre polynomial of degree 4.

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A384588 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384586.

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A384590 a(n) = floor(X(n,n)), where X(n,n) is the largest zero of the Laguerre polynomial of degree n.

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