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A094874 Decimal expansion of (5-sqrt(5))/2. 11

%I

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

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

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

%N Decimal expansion of (5-sqrt(5))/2.

%C Also the limiting ratio of Lucas(n)/Fibonacci(n+1), or Fibonacci(n-1)/Fibonacci(n+1) + 1. - _Alexander Adamchuk_, Oct 10 2007

%H Ivan Panchenko, <a href="/A094874/b094874.txt">Table of n, a(n) for n = 1..1000</a>

%H Paul Cooijmans, <a href="http://web.archive.org/web/20050302174449/http://members.chello.nl/p.cooijmans/gliaweb/tests/odds.html">Odds</a>.

%H Yiyan Ni, Myron Hlynka, Percy H. Brill, <a href="https://arxiv.org/abs/1806.09150">Urn Models and Fibonacci Series</a>, arXiv:1806.09150 [math.CO], 2018. See (9) p. 7.

%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.

%F (2-phi)*(2+phi) = 2 - 1/phi = 3 - phi = (5-sqrt(5))/2 = (2*sin(Pi/5))^2, where phi is the golden ratio (A001622).

%F Equals prod(n > 0, (1 + 1/A192223(n))). - _Charles R Greathouse IV_, Jun 26 2011

%F Equals Prod_{k >= 2} (-1)^k/(Fibonacci(k)*Fibonacci(k+1)). See Ni et al. - _Michel Marcus_, Jun 26 2018

%e 1.38196601125010515179541316563436188...

%t RealDigits[5/2 - Sqrt[5]/2, 10, 100][[1]] (* _Alonso del Arte_, Jun 26 2018 *)

%o (PARI) (5-sqrt(5))/2 \\ _Charles R Greathouse IV_, Jun 26 2011

%Y Equals A079585-1.

%Y Cf. A000032, A000045, A192223.

%K cons,nonn,easy

%O 1,2

%A _N. J. A. Sloane_, Jun 14 2004

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Last modified November 19 09:15 EST 2018. Contains 317348 sequences. (Running on oeis4.)