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

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

1,1

Links

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.

Formula

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

Example

4.53662029692112798327928538495713788012578435338680...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A275275 A376180 A387102 * A196402 A268741 A263031

Adjacent sequences: A384583 A384584 A384585 * A384587 A384588 A384589

Keyword

nonn,cons

Author

A.H.M. Smeets, Jun 04 2025

Status

approved