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Rounded total surface area of a regular tetrahedron with edge length n.
9

%I #33 Jun 20 2024 11:08:20

%S 0,2,7,16,28,43,62,85,111,140,173,210,249,293,339,390,443,501,561,625,

%T 693,764,838,916,998,1083,1171,1263,1358,1457,1559,1665,1774,1886,

%U 2002,2122,2245,2371,2501,2634,2771,2912,3055,3203,3353,3507,3665,3826,3991

%N Rounded total surface area of a regular tetrahedron with edge length n.

%C a(n) is the integer k that minimizes |k/n^2 - sqrt(3)|. - _Clark Kimberling_, Oct 11 2017

%D S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, pp. 10-11.

%H Vincenzo Librandi, <a href="/A070169/b070169.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tetrahedron.html">Tetrahedron</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlatonicSolid.html">Platonic Solid</a>

%F a(n) = round(n^2 * sqrt(3)).

%F a(n) = A000194(3*n^4). - _Christian Krause_, Aug 04 2021; corrected by _Chai Wah Wu_, Jun 19 2024

%e a(3)=16 because round(3^2*sqrt(3)) = round(9*1.73205...) = round(15.5884...) = 16.

%t Round[Sqrt[3]#]&/@(Range[0,50]^2) (* _Harvey P. Dale_, Sep 24 2012 *)

%o (PARI) for(n=0,100,print1(round(n^2*sqrt(3)),","))

%o (Magma) [Round(n^2 * Sqrt(3)): n in [0..50]]; // _Vincenzo Librandi_, May 21 2011

%o (Python)

%o from math import isqrt

%o def A070169(n): return (m:=isqrt(k:=3*n**4))+(k-m*(m+1)>=1) # _Chai Wah Wu_, Jun 19 2024

%Y Cf. A033581 (cube), A071396 (octahedron), A071397 (dodecahedron), A071398 (icosahedron), A071399 (volume of tetrahedron).

%K easy,nonn

%O 0,2

%A _Rick L. Shepherd_, Apr 24 2002