A382105 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.

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

Offset

0,1

Comments

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:

k | zeros | corresponding weights

---+---------------------------+--------------------------

2 | A020760 | A000007*10

3 | A010513/10 | A010716

4 | A372267, A372268 | A382103, A382104

5 | A372269, A372270 | this sequence, 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) Table 25.4, n=5.

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 (322+13*sqrt(70))/900.

Example

0.47862867049936646804129151483563819291229555334...

Mathematica

First[RealDigits[(322 + 13*Sqrt[70])/9, 10, 100]] (* Paolo Xausa, Nov 25 2025 *)

Prog

(PARI) (322+13*sqrt(70))/900 \\ Charles R Greathouse IV, Nov 24 2025

Crossrefs

Cf. A372269.

Sequence in context: A115021 A200367 A394607 * A272490 A261654 A332504

Adjacent sequences: A382102 A382103 A382104 * A382106 A382107 A382108

Keyword

nonn,cons,easy

Author

A.H.M. Smeets, Mar 27 2025

Status

approved