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%I #56 Apr 12 2021 10:52:44
%S 5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Decimal expansion of 1/2.
%C Real part of all nontrivial zeros of the Riemann zeta function (assuming the Riemann hypothesis to be true). - _Alonso del Arte_, Jul 02 2011
%C Radius of a sphere with surface area Pi. - _Omar E. Pol_, Aug 09 2012
%C Radius of the midsphere (tangent to the edges) in a regular octahedron with unit edges. Also radius of the inscribed sphere (tangent to faces) in a cube with unit edges. - _Stanislav Sykora_, Mar 27 2014
%C Construct a rectangle of maximal area inside an arbitrary triangle. The ratio of the rectangle's area to the triangle's area is 1/2. - _Rick L. Shepherd_, Jul 30 2014
%H Michael Penn, <a href="https://www.youtube.com/watch?v=hJzIZNvscpM">A creative approach to a scary looking integral.</a>, YouTube video, 2020.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=IXQNHqr1soo">I really like this sum!</a>, YouTube video, 2021.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Platonic solid">Platonic solid</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F Equals Sum_{k>=1} (1/3^k). Hence 1/2 = 0.1111111111111... in base 3.
%F Cosine of 60 degrees, i.e., cos(Pi/3).
%F -zeta(0), zeta being the Riemann function. - _Stanislav Sykora_, Mar 27 2014
%F a(0) = 5; a(n) = 0, n > 0. - _Wesley Ivan Hurt_, Mar 27 2014
%F a(n) = 5 * floor(1/(n + 1)). - _Wesley Ivan Hurt_, Mar 27 2014
%F Equals 2*A019824*A019884. - _R. J. Mathar_, Jan 17 2021
%e 1/2 = 0.50000000000000...
%p Digits:=100; evalf(1/2); # _Wesley Ivan Hurt_, Mar 27 2014
%t RealDigits[1/2, 10, 128][[1]] (* _Alonso del Arte_, Dec 13 2013 *)
%t LinearRecurrence[{1},{5,0},99] (* _Ray Chandler_, Jul 15 2015 *)
%o (PARI) { default(realprecision); x=1/2*10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ _Felix Fröhlich_, Jul 24 2014
%o (PARI) a(n) = 5*(n==0); \\ _Michel Marcus_, Jul 25 2014
%Y Cf. In platonic solids:
%Y midsphere radii:
%Y A020765 (tetrahedron),
%Y A010503 (cube),
%Y A019863 (icosahedron),
%Y A239798 (dodecahedron);
%Y insphere radii:
%Y A020781 (tetrahedron),
%Y A020763 (octahedron),
%Y A179294 (icosahedron),
%Y A237603 (dodecahedron).
%K nonn,cons,easy
%O 0,1
%A _N. J. A. Sloane_
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