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

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

0,1

Comments

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

Links

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.

Index entries for algebraic numbers, degree 2.

Formula

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

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

Example

0.65214515486254614262693605077800059276465130416610645...

Mathematica

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

Prog

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

Crossrefs

Cf. A372268.

Sequence in context: A177938 A374529 A112282 * A098866 A144689 A221215

Adjacent sequences: A382101 A382102 A382103 * A382105 A382106 A382107

Keyword

nonn,cons

Author

A.H.M. Smeets, Mar 15 2025

Status

approved