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Rounded volume of a regular icosahedron with edge length n.
6

%I #14 Sep 08 2022 08:45:06

%S 0,2,17,59,140,273,471,748,1117,1590,2182,2904,3770,4793,5987,7363,

%T 8936,10719,12724,14964,17454,20205,23231,26545,30160,34089,38345,

%U 42942,47893,53209,58906,64995,71490,78404,85749,93540,101789,110509

%N Rounded volume of a regular icosahedron with edge length n.

%C The printed reference given shows in a table on p. 10 that Volume is "2.18170a^3" (a is edge). Both PARI (see Example here) and a handheld calculator show that 2.18169 is correct for a 5-decimal-place approximation.

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

%H Vincenzo Librandi, <a href="/A071402/b071402.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) = round(n^3 * (3+sqrt(5)) * 5/12).

%e a(6)=471 because round(6^3*(3 + sqrt(5))*5/12) = round(216*2.181694990...) = round(471.24...) = 471.

%o (PARI) for(n=0,100,print1(round(n^3*(3+sqrt(5))*5/12),","))

%o (Magma) [Round(n^3 * (3+Sqrt(5)) * 5/12): n in [0..50]]; // _Vincenzo Librandi_, May 21 2011

%Y Cf. A000578 (cube), A071399 (tetrahedron), A071400 (octahedron), A071401 (dodecahedron), A071398 (total surface area of icosahedron).

%Y Cf. A102208 ((3+Sqrt(5)) * 5/12).

%K easy,nonn

%O 0,2

%A _Rick L. Shepherd_, May 29 2002