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 A109134 Decimal expansion of Phi, the real root of the equation 1/x = (x-1)^2. 7

%I

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

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

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

%N Decimal expansion of Phi, the real root of the equation 1/x = (x-1)^2.

%C The silver number (A060006) is equal to Phi*(Phi-1).

%C Also Phi*(Phi-1) = 1/(Phi-1). - _Richard R. Forberg_, Oct 08 2014

%C Equations to which this is a root can also be written as: x = sqrt(x + sqrt(x)); x^2 - x - sqrt(x) = 0; or this form where n = 1: x = n + 1/sqrt(x). When n = 2 then the root is 2.618033988... = A104457 = 1 + A001622 or 1 + "Golden Ratio" called phi. - _Richard R. Forberg_, Oct 08 2014

%C Also equals the largest root (negated) of the Mandelbrot polynomial P_2(z) = 1+z*(1+z)^2. - _Jean-François Alcover_, Apr 16 2015

%C Suppose that r is a real number in the interval [3/2, 5/3). Let C(r) = (c(k)) be the sequence of coefficients in the Maclaurin series for 1/(Sum_{k>=0} floor((k+1)*r))(-x)^k). Conjectures: the limit L(r) of c(k+1)/c(k) as k -> oo exists, L(r) is discontinuous at 5/3 (cf. A279676), and the left limit of L(r) as r->5/3 is Phi. - _Clark Kimberling_, Jul 11 2017

%D M. Gardner, A Gardner's Workout, pp. 124-126, A. K. Peters MA 2001.

%H Vincenzo Librandi, <a href="/A109134/b109134.txt">Table of n, a(n) for n = 1..5000</a>

%H Simon Baker, <a href="https://doi.org/10.1016/j.jnt.2014.08.003">On small bases which admit countably many expansions</a>, Journal of Number Theory, Volume 147, February 2015, Pages 515-532.

%H Simon Plouffe, <a href="http://pi.lacim.uqam.ca/eng/">Plouffe's Inverter</a>.

%H Nikita Sidorov, <a href="https://doi.org/10.1016/j.jnt.2008.11.003">Expansions in non-integer bases: Lower, middle and top orders</a>, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Prop. 2.3 p. 744.

%H Yuru Zou, Derong Kong, <a href="https://doi.org/10.1016/j.jnt.2015.06.017">On a problem of countable expansions</a>, Journal of Number Theory, Volume 158, January 2016, Pages 134-150. See Theorem 1.1 p. 135.

%F Equals 1+A075778. - _R. J. Mathar_, Aug 20 2008

%F Equals (1/6*(108+12*sqrt(69))^(1/3) + 2/(108+12*sqrt(69))^(1/3))^2. - _Vaclav Kotesovec_, Oct 08 2014

%e 1.75487766624669276004950889635852869189460661777279314398928397064...

%t FindRoot[x^3 - 2x^2 + x - 1 == 0, {x, 1.75}, WorkingPrecision -> 128][[1, 2]] (* _Robert G. Wilson v_, Aug 19 2005 *)

%t Root[x^3-2x^2+x-1, x, 1] // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Mar 05 2013 *)

%o (PARI) {d=104;default(realprecision,d);print(k=solve(x=1,2,(x-1)^2-1/x)); for(c=0,d,z=floor(k);print1(z,",",);k=10*(k-z))}

%o (PARI) polrootsreal(x^3-2*x^2+x-1)[1] \\ _Charles R Greathouse IV_, Aug 15 2014

%Y Cf. A001622, A001685, A060006, A075778, A104457.

%K cons,nonn

%O 1,2

%A _Lekraj Beedassy_, Aug 17 2005

%E Extended by _Klaus Brockhaus_ and _Robert G. Wilson v_, Aug 19 2005

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Last modified May 31 16:29 EDT 2020. Contains 334748 sequences. (Running on oeis4.)