A393362 Decimal expansion of the smallest positive zero of the Hermite polynomial of degree 6.

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

Offset

0,1

Comments

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

k | zeros | corresponding weights for Hermite-Gauss quadrature

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

2 | A010503 | A019704

3 | 0, A115754 | 10*A019717, A019708

4 | A393353, A393354 | A393355, A393356

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

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

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

Links

A.H.M. Smeets, 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]. Eq. 25.4.46 p. 890 and Table 25.10 p. 924.

Eric Weisstein's World of Mathematics, Hermite-Gauss Quadrature.

Index entries for algebraic numbers, degree 6.

Formula

Minimal polynomial 8*x^6 -60*x^4 +90*x^2 -15. - R. J. Mathar, Jun 03 2026

Example

0.43607741192761650867921594825062491076625666893...

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A257301 A298801 A340297 * A394652 A369928 A016503

Adjacent sequences: A393359 A393360 A393361 * A393363 A393364 A393365

Keyword

nonn,cons

Author

A.H.M. Smeets, Feb 24 2026

Status

approved