A384590 a(n) = floor(X(n,n)), where X(n,n) is the largest zero of the Laguerre polynomial of degree n.

1, 3, 6, 9, 12, 15, 19, 22, 26, 29, 33, 37, 40, 44, 48, 51, 55, 59, 62, 66, 70, 73, 77, 81, 85, 89, 92, 96, 100, 104, 107, 111, 115, 119, 123, 126, 130, 134, 138, 142, 146, 149, 153, 157, 161, 165, 169, 172, 176, 180, 184, 188, 192, 196, 199, 203, 207, 211

Offset

1,2

Comments

For X(k,n), the k-th smallest zero of the Laguerre polynomial of degree n, see formula section of A091476, for large n and relative small k, k << n.

Some terms for large n:

a(1000) = floor(3943.2473948452...), a(2000) = floor(7927.9014222639...), a(4000) = floor(15908.5812117320...), a(8000) = floor(31884.2511300626...), a(16000) = floor(63853.6067816122...), a(32000) = floor(127815.0051094389...), a(64000) = floor(255766.3763209512...), a(128000) = floor(511705.1129209706...), a(256000) = floor(1023627.9299056501...), a(512000) = floor(2047530.6886230061...).

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..12245

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], Table 25.9.

Luigi Gatteschi, Asymptotics and bounds for the zeros of Laguerre polynomials: A survey, J. Comput. Appl. Math. 144 (1-2), 2002, pp. 7-27.

Vaclav Kotesovec, Graph - the asymptotic ratio

A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.

Francesco Tricomi, Sul comportamento asintotico dell'n-esimo polinomio di Laguerre nell'intorno dell'ascissa 4n, Comment. Math Helv. 22 (1949) 150-167.

Eric Weisstein's World of Mathematics, Laguerre Polynomial.

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

Formula

Limit_{n -> oo} X(n,n)/n = 4.

a(n) ~ floor(4*n + 2 - 5.8917*n^(1/3)).

The constant above is equal to -5.89166148670690567478546401633382049 = c * 2^(4/3), where c = -2.338107410459767... is the first negative zero of the Airy function Ai(c). [Tricomi, 1949] - Vaclav Kotesovec, Jan 16 2026

Mathematica

A384590[n_] := Floor[Root[LaguerreL[n, #] &, n]];
Array[A384590, 70] (* Paolo Xausa, Jun 26 2025 *)

Crossrefs

Cf. A091476.

Cf. 1+A014176 (n=2), A384279 (n=3), A384587 (n=4).

Sequence in context: A083354 A392352 A156242 * A060293 A336803 A220657

Adjacent sequences: A384587 A384588 A384589 * A384591 A384592 A384593

Keyword

nonn

Author

A.H.M. Smeets, Jun 14 2025

Status

approved