A393362 - OEIS

%I #13 Jun 03 2026 09:46:53

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

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

%U 3,2,1,7,1,6,5,6,2,1,7,2,9,0,2,9,0,4,3,3

%N Decimal expansion of the smallest positive zero of the Hermite polynomial of degree 6.

%C There are floor(k/2) positive zeros of the Hermite polynomial of degree k:

%C    k | zeros                           | corresponding weights for Hermite-Gauss quadrature

%C   ---+---------------------------------+----------------------------------------------------

%C    2 | A010503                         | A019704

%C    3 | 0, A115754                      | 10*A019717, A019708

%C    4 | A393353, A393354                | A393355, A393356

%C    5 | 0, A393357, A393358             | A245887, A393360, A393361

%C    6 | this sequence, A393363, A393364 | A393365, A393366, A393367

%C    7 | 0, A393368, A393369, A393370    | A393371, A393372, A393373, A393374

%H A.H.M. Smeets, <a href="/A393362/b393362.txt">Table of n, a(n) for n = 0..10000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Eq. 25.4.46 p. 890 and Table 25.10 p. 924.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hermite-GaussQuadrature.html">Hermite-Gauss Quadrature</a>.

%H <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>.

%F Minimal polynomial 8*x^6 -60*x^4 +90*x^2 -15. - _R. J. Mathar_, Jun 03 2026

%e 0.43607741192761650867921594825062491076625666893...

%t RealDigits[x /. FindRoot[HermiteH[6, x], {x, 1/2}, WorkingPrecision -> 100]][[1]] (* _Amiram Eldar_, Feb 24 2026 *)

%o (PARI) polrootsreal(polhermite(6))[4]

%K nonn,cons

%O 0,1

%A _A.H.M. Smeets_, Feb 24 2026