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

Author

A.H.M. Smeets, Mar 15 2025

Keywords

• nonn
• cons

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

Examples

			0.65214515486254614262693605077800059276465130416610645...
		

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.

Crossrefs

Cf. A372268.

Programs

Mathematica

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

PARI

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

Formula

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

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