A372267 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.

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

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 25.4, n=4

Wikipedia, Legendre polynomials.

Index entries for algebraic numbers, degree 4.

Formula

Smallest positive root of 35*x^4 - 30*x^2 + 3 = 0.

Equals sqrt((3-2*sqrt(6/5))/7).

Example

0.339981043584856264802665759103244687200575869770914352592953...

Mathematica

First[RealDigits[Root[LegendreP[4, #] &, 3], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Crossrefs

Cf. A008316, A100258.

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

k | zeros

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

2 | A020760

3 | A010513/10

4 | A372267, A372268

5 | A372269, A372270

6 | A372271, A372272, A372273

7 | A372274, A372275, A372276

Sequence in context: A388529 A201456 A392688 * A347033 A336992 A349673

Adjacent sequences: A372264 A372265 A372266 * A372268 A372269 A372270

Keyword

nonn,cons

Author

Pontus von Brömssen, Apr 25 2024

Status

approved