A393369 Decimal expansion of the second smallest positive zero of the Hermite polynomial of degree 7.

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

Offset

1,2

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 | A393362, A393363, A393364 | A393365, A393366, A393367

7 | 0, A393368, this sequence, A393370 | A393371, A393372, A393373, A393374

Links

Table of n, a(n) for n=1..88.

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 -84*x^4 +210*x^2 -105. - R. J. Mathar, Jun 03 2026

Example

1.6735516287674714450318013983035948191078100577...

Mathematica

RealDigits[x /. FindRoot[HermiteH[7, x], {x, 2}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Mar 02 2026 *)

Prog

(PARI) polrootsreal(polhermite(7))[6]

Crossrefs

Cf. A060821.

Sequence in context: A105739 A105831 A248650 * A154339 A382235 A197141

Adjacent sequences: A393366 A393367 A393368 * A393370 A393371 A393372

Keyword

nonn,cons

Author

A.H.M. Smeets, Mar 02 2026

Status

approved