A372268 - OEIS

A372268

Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.

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

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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

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

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

EXAMPLE

0.861136311594052575223946488892809505095725379629717637615721...

MATHEMATICA

First[RealDigits[Root[LegendreP[4, #] &, 4], 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: