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A171972 Greatest integer k such that k/n^2 < sqrt(3).

%I A171972 #16 Feb 16 2025 08:33:11
%S A171972 0,1,6,15,27,43,62,84,110,140,173,209,249,292,339,389,443,500,561,625,
%T A171972 692,763,838,916,997,1082,1170,1262,1357,1456,1558,1664,1773,1886,
%U A171972 2002,2121,2244,2371,2501,2634,2771,2911,3055,3202,3353,3507,3665,3826,3990,4158
%N A171972 Greatest integer k such that k/n^2 < sqrt(3).
%C A171972 Integer part of the surface area of a regular tetrahedron with edge length n.
%C A171972 A171970(n)*A005843(n) <= a(n);
%C A171972 a(n) <= 4*A171971(n); 0 <= a(n) - 4*A171971(n) < 4.
%H A171972 Reinhard Zumkeller, <a href="/A171972/b171972.txt">Table of n, a(n) for n = 0..1000</a>
%H A171972 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Tetrahedron.html">Tetrahedron</a>
%H A171972 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedron">Tetrahedron</a>
%F A171972 a(n) = floor(n^2 * sqrt(3)).
%F A171972 a(n) = A022838(n^2);
%F A171972 a(n) = A293410(n) - 1 for n > 0.
%t A171972 z = 120; r = Sqrt[3];
%t A171972 Table[Floor[r*n^2], {n, 0, z}]; (* A171972 *)
%t A171972 Table[Ceiling[r*n^2], {n, 0, z}]; (* A293410 *)
%t A171972 Table[Round[r*n^2], {n, 0, z}]; (* A070169. -  _Clark Kimberling_, Oct 11 2017 *)
%o A171972 (Haskell)
%o A171972 a171972 = floor . (* sqrt 3) . fromInteger . a000290
%o A171972 -- _Reinhard Zumkeller_, Dec 15 2012
%Y A171972 Cf. A002194, A070169, A171974, A171973, A171975, A000290, A293410.
%K A171972 nonn
%O A171972 0,3
%A A171972 _Reinhard Zumkeller_, Jan 20 2010