A204188 - OEIS

%I #26 Dec 04 2018 07:42:12

%S 5,5,9,0,1,6,9,9,4,3,7,4,9,4,7,4,2,4,1,0,2,2,9,3,4,1,7,1,8,2,8,1,9,0,

%T 5,8,8,6,0,1,5,4,5,8,9,9,0,2,8,8,1,4,3,1,0,6,7,7,2,4,3,1,1,3,5,2,6,3,

%U 0,2,3,1,4,0,9,4,5,1,2,2,4,8,5,3,6,0,3,6,0,2,0,9,4,6,9,5,5,6,8,7

%N Decimal expansion of sqrt(5)/4.

%C Equals Product_{n>=1} (1 - 1/A000032(2^n)).

%C Essentially the same as A019863 and A019827. - _R. J. Mathar_, Jan 16 2012

%C The value is the distance of the W point of the Wigner-Seitz cell of the body-centered cubic lattice (that is the Brioullin zone of the face-centered cubic lattice) to its four nearest neighbors. Let the points of the simple cubic lattice be at (1,0,0), (0,1,0), (1,0,0) etc and the point in the cube center at (1/2, 1/2, 1/2). Then W is at (0, 1/4, 1/2) [or any of the 24 symmetry related positions like (0, 3/4, 1/2), (0, 1/2, 1/4) etc.], and the four lattice points closest to W are at (-1/2, 1/2, 1/2), (0,0,0), (1/2, 1/2, 1/2) and (0,0,1). - _R. J. Mathar_, Aug 19 2013

%H J. Sondow, <a href="http://arxiv.org/abs/1106.4246">Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers</a>, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.

%H Y. Tachiya, <a href="http://dx.doi.org/10.1016/j.jnt.2006.11.006">Transcendence of certain infinite products</a>, J. Number Theory 125 (2007), 182-200.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Brillouin_zone">Brillouin zone</a>

%F Equals sqrt(5)/4 = (-1 + 2*phi)/4, with the golden section phi from A001622.

%F Equals 5*A020837.

%e 0.5590169943749474241022934171828190588601545899028814310677243113526302...

%p evalf(sqrt(5)/4);

%t RealDigits[Sqrt[5]/4, 10, 100][[1]] (* _Amiram Eldar_, Dec 04 2018 *)

%o (PARI) sqrt(5)/4 \\ _Charles R Greathouse IV_, Apr 21 2016

%Y Cf. A001622, A002163.

%K nonn,cons

%O 0,1

%A _Jonathan Sondow_, Jan 14 2012