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A372269 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 5.

%I A372269 #14 Feb 27 2025 04:24:59
%S A372269 5,3,8,4,6,9,3,1,0,1,0,5,6,8,3,0,9,1,0,3,6,3,1,4,4,2,0,7,0,0,2,0,8,8,
%T A372269 0,4,9,6,7,2,8,6,6,0,6,9,0,5,5,5,9,9,5,6,2,0,2,2,3,1,6,2,7,0,5,9,4,7,
%U A372269 1,1,8,5,3,6,7,7,5,5,2,9,1,0,3,5,8,0,3
%N A372269 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 5.
%H A372269 Paolo Xausa, <a href="/A372269/b372269.txt">Table of n, a(n) for n = 0..10000</a>
%H A372269 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_polynomials">Legendre polynomials</a>.
%H A372269 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A372269 Smallest positive root of 63*x^4 - 70*x^2 + 15 = 0.
%F A372269 Equals sqrt(5-2*sqrt(10/7))/3.
%e A372269 0.538469310105683091036314420700208804967286606905559956202231...
%t A372269 First[RealDigits[Root[LegendreP[5, #] &, 4], 10, 100]] (* _Paolo Xausa_, Feb 27 2025 *)
%Y A372269 Cf. A008316, A100258.
%Y A372269 There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
%Y A372269    k | zeros
%Y A372269   ---+--------------------------
%Y A372269    2 | A020760
%Y A372269    3 | A010513/10
%Y A372269    4 | A372267, A372268
%Y A372269    5 | A372269, A372270
%Y A372269    6 | A372271, A372272, A372273
%Y A372269    7 | A372274, A372275, A372276
%K A372269 nonn,cons
%O A372269 0,1
%A A372269 _Pontus von Brömssen_, Apr 25 2024