A377805 - OEIS

%I #18 Feb 11 2025 09:55:10

%S 3,7,6,1,6,6,4,9,9,6,2,7,3,3,3,6,2,9,7,5,7,7,7,6,7,3,6,7,1,3,0,2,7,1,

%T 4,3,4,0,3,5,5,2,8,9,8,7,3,4,8,8,0,9,8,9,6,0,4,9,6,8,9,7,3,0,2,9,9,3,

%U 6,2,0,0,7,5,7,8,7,6,4,1,6,7,9,4,6,0,9,2,9,4

%N Decimal expansion of the volume of a snub dodecahedron with unit edge length.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SnubDodecahedron.html">Snub Dodecahedron</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Snub_dodecahedron">Snub dodecahedron</a>.

%H <a href="/index/Al#algebraic_12">Index entries for algebraic numbers, degree 12</a>.

%F Equals ((3*phi + 1)*xi*(xi + 1) - phi/6 - 2)/sqrt(3*xi^2 - phi^2) = (A090550*xi*(xi + 1) - A134946 - 2)/sqrt(3*xi^2 - A104457), where phi = A001622 and xi = A377849.

%F Equals the real root closest to 37 of 2176782336*x^12 - 3195335070720*x^10 + 162223191936000*x^8 + 1030526618040000*x^6 + 6152923794150000*x^4 - 182124351550575000*x^2 + 187445810737515625.

%e 37.616649962733362975777673671302714340355289873...

%t First[RealDigits[((3*GoldenRatio + 1)*#*(# + 1) - GoldenRatio/6 - 2)/Sqrt[3*#^2 - GoldenRatio^2], 10, 100]] & [Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1]] (* or *)

%t First[RealDigits[PolyhedronData["SnubDodecahedron", "Volume"], 10, 100]]

%Y Cf. A377804 (surface area), A377806 (circumradius), A377807 (midradius).

%Y Cf. A102769 (analogous for a regular dodecahedron).

%Y Cf. A001622, A090550, A104457, A134946, A377849.

%K nonn,cons,easy

%O 2,1

%A _Paolo Xausa_, Nov 09 2024