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

Author

A.H.M. Smeets, Jun 14 2025

Keywords

• nonn

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.
• A.H.M. Smeets, Abscissas and weight factors for Laguerre integration for some larger degrees.
• Eric Weisstein's World of Mathematics, Laguerre Polynomial.
• Eric Weisstein's World of Mathematics, Laguerre-Gauss Quadrature.

Crossrefs

Cf. A091476.

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

Programs

Mathematica

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

Formula

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

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