A070169 Rounded total surface area of a regular tetrahedron with edge length n.

0, 2, 7, 16, 28, 43, 62, 85, 111, 140, 173, 210, 249, 293, 339, 390, 443, 501, 561, 625, 693, 764, 838, 916, 998, 1083, 1171, 1263, 1358, 1457, 1559, 1665, 1774, 1886, 2002, 2122, 2245, 2371, 2501, 2634, 2771, 2912, 3055, 3203, 3353, 3507, 3665, 3826, 3991

Offset: 0

Author

Rick L. Shepherd, Apr 24 2002

Keywords

• easy
• nonn

Comments

a(n) is the integer k that minimizes |k/n^2 - sqrt(3)|. - Clark Kimberling, Oct 11 2017

Examples

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

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Tetrahedron
• Eric Weisstein's World of Mathematics, Platonic Solid

Crossrefs

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

Programs

Magma

[Round(n^2 * Sqrt(3)): n in [0..50]]; // Vincenzo Librandi, May 21 2011

Mathematica

Round[Sqrt[3]#]&/@(Range[0,50]^2) (* Harvey P. Dale, Sep 24 2012 *)

PARI

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

Python

from math import isqrt
def A070169(n): return (m:=isqrt(k:=3*n**4))+(k-m*(m+1)>=1) # Chai Wah Wu, Jun 19 2024

Formula

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

a(n) = A000194(3*n^4). - Christian Krause, Aug 04 2021; corrected by Chai Wah Wu, Jun 19 2024