

A109134


Decimal expansion of Phi, the real root of the equation 1/x = (x1)^2.


8



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, 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, 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
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OFFSET

1,2


COMMENTS

The silver number (A060006) is equal to Phi*(Phi1).
Also Phi*(Phi1) = 1/(Phi1).  Richard R. Forberg, Oct 08 2014
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
Also equals the largest root (negated) of the Mandelbrot polynomial P_2(z) = 1+z*(1+z)^2.  JeanFrançois Alcover, Apr 16 2015
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


REFERENCES

M. Gardner, A Gardner's Workout, pp. 124126, A. K. Peters MA 2001.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Simon Baker, On small bases which admit countably many expansions, Journal of Number Theory, Volume 147, February 2015, Pages 515532.
Simon Plouffe, Plouffe's Inverter .
Nikita Sidorov, Expansions in noninteger bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741754. See Prop. 2.3 p. 744.
Yuru Zou, Derong Kong, On a problem of countable expansions, Journal of Number Theory, Volume 158, January 2016, Pages 134150. See Theorem 1.1 p. 135.


FORMULA

Equals 1+A075778.  R. J. Mathar, Aug 20 2008
Equals (1/6*(108+12*sqrt(69))^(1/3) + 2/(108+12*sqrt(69))^(1/3))^2.  Vaclav Kotesovec, Oct 08 2014
Equals Rho^2 where Rho is the plastic number 1.3247179572...(see A060006).  Philippe Deléham, Sep 29 2020


EXAMPLE

1.75487766624669276004950889635852869189460661777279314398928397064...


MATHEMATICA

FindRoot[x^3  2x^2 + x  1 == 0, {x, 1.75}, WorkingPrecision > 128][[1, 2]] (* Robert G. Wilson v, Aug 19 2005 *)
Root[x^32x^2+x1, x, 1] // RealDigits[#, 10, 105]& // First (* JeanFrançois Alcover, Mar 05 2013 *)


PROG

(PARI) {d=104; default(realprecision, d); print(k=solve(x=1, 2, (x1)^21/x)); for(c=0, d, z=floor(k); print1(z, ", ", ); k=10*(kz))}
(PARI) polrootsreal(x^32*x^2+x1)[1] \\ Charles R Greathouse IV, Aug 15 2014


CROSSREFS

Cf. A001622, A001685, A060006, A075778, A104457.
Sequence in context: A289032 A289005 A225408 * A075778 A289033 A010510
Adjacent sequences: A109131 A109132 A109133 * A109135 A109136 A109137


KEYWORD

cons,nonn


AUTHOR

Lekraj Beedassy, Aug 17 2005


EXTENSIONS

Extended by Klaus Brockhaus and Robert G. Wilson v, Aug 19 2005


STATUS

approved



