A393353 Decimal expansion of the smallest positive zero of the Hermite polynomial of degree 4.

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

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 | this sequence, A393354 | A393355, A393355

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

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

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

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]. 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 4.

Formula

Equals sqrt((3-sqrt(6))/2).

Smallest positive real root of 4*x^4 - 12*x^2 + 3 = 0 (see also A060821).

Example

0.5246476232752903178840602538347413414135785651694633...

Mathematica

First[RealDigits[Root[HermiteH[4, #] &, 3], 10, 100]] (* Paolo Xausa, Feb 19 2026 *)

Prog

(PARI) polrootsreal(polhermite(4))[3]

Crossrefs

Cf. A060821.

Sequence in context: A188739 A308171 A376234 * A265287 A329477 A257701

Adjacent sequences: A393350 A393351 A393352 * A393354 A393355 A393356

Keyword

nonn,cons

Author

A.H.M. Smeets, Feb 12 2026

Status

approved