A372276 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.

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

Offset: 0

Author

Pontus von Brömssen, Apr 25 2024

Keywords

• nonn
• cons

Examples

			0.949107912342758524526189684047851262400770937670617783548769...
		

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 25.4, n=7.
• A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
• Eric Weisstein's World of Mathematics, Legendre Polynomial.
• Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
• Wikipedia, Legendre polynomials.
• Index entries for algebraic numbers, degree 6.

Crossrefs

Cf. A008316, A100258.

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, A382104

5 | A372269, A372270 | A382105, A382106

6 | A372271, A372272, A372273 | A382107, A382686, A382687

7 | A372274, A372275, this sequence | A382688, A382689, A382690

Programs

Mathematica

First[RealDigits[Root[LegendreP[7, #] &, 7], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

PARI

solve (x = 0.8, 1.0, 429*x^6 - 693*x^4 + 315*x^ - 35) \\ A.H.M. Smeets, May 31 2025

Formula

Largest positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.