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A192918 The decimal expansion of the real root of r^3 + r^2 + r - 1. 12
5, 4, 3, 6, 8, 9, 0, 1, 2, 6, 9, 2, 0, 7, 6, 3, 6, 1, 5, 7, 0, 8, 5, 5, 9, 7, 1, 8, 0, 1, 7, 4, 7, 9, 8, 6, 5, 2, 5, 2, 0, 3, 2, 9, 7, 6, 5, 0, 9, 8, 3, 9, 3, 5, 2, 4, 0, 8, 0, 4, 0, 3, 7, 8, 3, 1, 1, 6, 8, 6, 7, 3, 9, 2, 7, 9, 7, 3, 8, 6, 6, 4, 8, 5, 1, 5, 7, 9, 1, 4, 5, 7, 6, 0, 5, 9, 1, 2, 5, 4, 6, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The real root r of the cubic equation r^3 + r^2 + r - 1 = 0 is the reciprocal of the tribonacci constant A058265. If the four sides of a quadrilateral form a geometric progression 1:r:r^2:r^3 where r is the common ratio then r is limited to the range 1/t < r < t where t is the tribonacci constant. More generally if f(n) is the n-th step Fibonacci constant then a polygon of n+1 sides can have sides in a geometric progression 1:r:r^2:....:r^n if the common ratio r is limited to the range 1/f(n) < r < f(n).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

Wikipedia, Triangle whose sides form a geometric progression.

FORMULA

r = (1/3)*(-1-2/(17+3*sqrt(33))^(1/3)+(17+3*sqrt(33))^(1/3)).

EXAMPLE

0.543689012692076361570855971801747986525203297650983935240...

MATHEMATICA

N[Reduce[r+r^2+r^3==1, r], 100]

RealDigits[(1/3)*(-1 -2/(17+3*Sqrt[33])^(1/3) +(17+3*Sqrt[33])^(1/3)), 10, 100][[1]] (* G. C. Greubel, Feb 06 2019 *)

PROG

(PARI) polrootsreal(r^3 + r^2 + r - 1)[1] \\ Charles R Greathouse IV, Apr 14 2014

(MAGMA) SetDefaultRealField(RealField(100)); (1/3)*(-1 -2/(17 +3*Sqrt(33))^(1/3) +(17+3*Sqrt(33))^(1/3)); // G. C. Greubel, Feb 06 2019

(Sage) numerical_approx((1/3)*(-1 -2/(17+3*sqrt(33))^(1/3) +(17+ 3*sqrt(33))^(1/3)), digits=100) # G. C. Greubel, Feb 06 2019

CROSSREFS

Reciprocal of A058265.

Sequence in context: A019762 A328015 A256099 * A092156 A086409 A202412

Adjacent sequences:  A192915 A192916 A192917 * A192919 A192920 A192921

KEYWORD

nonn,cons

AUTHOR

Frank M Jackson, Aug 26 2011

STATUS

approved

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Last modified November 20 10:09 EST 2019. Contains 329334 sequences. (Running on oeis4.)