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

%I A384590 #25 Jul 08 2025 22:25:00
%S A384590 1,3,6,9,12,15,19,22,26,29,33,37,40,44,48,51,55,59,62,66,70,73,77,81,
%T A384590 85,89,92,96,100,104,107,111,115,119,123,126,130,134,138,142,146,149,
%U A384590 153,157,161,165,169,172,176,180,184,188,192,196,199,203,207,211
%N A384590 a(n) = floor(X(n,n)), where X(n,n) is the largest zero of the Laguerre polynomial of degree n.
%C A384590 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.
%C A384590 Some terms for large n:
%C A384590 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...).
%H A384590 A.H.M. Smeets, <a href="/A384590/b384590.txt">Table of n, a(n) for n = 1..12245</a>
%H A384590 Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], Table 25.9.
%H A384590 A.H.M. Smeets, <a href="/A384277/a384277.txt">Abscissas and weight factors for Laguerre integration for some larger degrees</a>.
%H A384590 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>.
%H A384590 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Laguerre-GaussQuadrature.html">Laguerre-Gauss Quadrature</a>.
%F A384590 Limit_{n -> oo} X(n,n)/n = 4.
%F A384590 a(n) ~ floor(4*n + 2 - 5.8917*n^(1/3)).
%t A384590 A384590[n_] := Floor[Root[LaguerreL[n, #] &, n]];
%t A384590 Array[A384590, 70] (* _Paolo Xausa_, Jun 26 2025 *)
%Y A384590 Cf. A091476.
%Y A384590 Cf. 1+A014176 (n=2), A384279 (n=3), A384587 (n=4).
%K A384590 nonn
%O A384590 1,2
%A A384590 _A.H.M. Smeets_, Jun 14 2025