A384463 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384277.

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

Offset

0,1

Links

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-Gauss Quadrature.

Index entries for algebraic numbers, degree 3.

Formula

Largest root of 1944*x^3 - 1944*x^2 + 405*x - 4 = 0.

Example

0.71109300992917301544959019114259443130939379628955...

Mathematica

First[RealDigits[Root[1944*#^3 - 1944*#^2 + 405*# - 4 &, 3], 10, 100]] (* Paolo Xausa, Jun 26 2025 *)

Prog

(PARI) solve(x = 0.7, 0.8, 1944*x^3 - 1944*x^2 + 405*x - 4)

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 | this sequence, A384464, A384465

4 | A384280, A384281 | A384466, A384467

Sequence in context: A370363 A249776 A348970 * A053878 A070672 A319101

Adjacent sequences: A384460 A384461 A384462 * A384464 A384465 A384466

Keyword

nonn,cons

Author

A.H.M. Smeets, May 30 2025

Status

approved