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%I #20 Feb 25 2023 11:45:52
%S 6,6,4,9,0,8,8,9,4,2,0,5,3,2,6,6,4,3,1,1,4,4,2,8,4,4,6,7,0,8,6,3,3,7,
%T 1,6,1,6,4,8,7,6,5,8,0,5,5,5,6,9,1,9,3,8,1,0,5,7,5,9,2,6,0,5,7,2,2,9,
%U 6,4,7,1,8,1,8,7,7,3,2,5,9,7,4,9,7,0,8,9,0,0,2,6,9,2,0,9,2,5,9,8,9,8,2,8,0
%N Decimal expansion of (5*sqrt(3) + sqrt(15))/(6*Pi).
%C The ratio of the volume of a regular dodecahedron to the volume of the circumscribed sphere (which has circumradius a*(sqrt(3) + sqrt(15))/4 = a*(A002194 + A010472)/4, where a is the dodecahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165954. A063723 shows the order of these by size.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dodecahedron.html">Dodecahedron</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F Equals (5*A002194 + A010472)/(6*A000796).
%F Equals (5*A002194 + A010472)*A049541/6.
%F Equals (10*A010527 + A010472)*A049541/6.
%F Equals (5 + sqrt(5))/(2*Pi*sqrt(3)).
%F Equals (5 + A002163)*A049541*A020760/2.
%e 0.6649088942053266431144284467086337161648765805556919381057592605722964718...
%t RealDigits[(5*Sqrt[3]+Sqrt[15])/(6*Pi),10,120][[1]] (* _Harvey P. Dale_, Feb 16 2018 *)
%o (PARI) (5*sqrt(3)+sqrt(15))/(6*Pi)
%Y Cf. A000796, A002194, A010472, A165922, A049541, A165952, A165954, A063723, A002163, A020760, A010527.
%K cons,nonn
%O 0,1
%A _Rick L. Shepherd_, Oct 02 2009
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