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A243508 Decimal expansion of the real positive root of 48x^4 + 16x^3 - 27x^2 - 18x - 3 = 0. 0
8, 8, 0, 5, 8, 3, 3, 4, 8, 3, 3, 9, 8, 2, 8, 1, 2, 4, 2, 1, 2, 9, 2, 3, 7, 8, 3, 7, 8, 4, 4, 9, 8, 7, 4, 3, 6, 8, 2, 4, 1, 8, 6, 4, 8, 4, 6, 8, 1, 5, 3, 1, 7, 1, 8, 1, 1, 0, 0, 1, 8, 1, 8, 6, 8, 5, 4, 4, 8, 4, 7, 7, 0, 5, 6, 8, 1, 6, 5, 2, 8, 3, 6, 5, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Given an equilateral triangle of unit length with two cevians drawn from one vertex to the side opposite that divide the equilateral triangle into 3 subtriangles. Adjust these cevians so that the 3 subtriangles all have congruent incircles. Then the real positive root of 48x^4 + 16x^3 - 27x^2 - 18x - 3 = 0 gives x = 1/4(3^(1/3)+3^(2/3)) = 0.880583348... as the length of these cevians and the radius of the three congruent incircles is given by A/(s+2x) where A is the area and s the semiperimeter of the equilateral triangle. Hence the congruent inradius = Sqrt(3)/(2(3+3^(2/3)+3^(1/3)).

A cubic number with denominator 4. - Charles R Greathouse IV, Aug 26 2017

LINKS

Table of n, a(n) for n=0..84.

Don McConnell, Three Congruent Incircles of a divided Equilateral triangle

Paul Yiu, The Congruent-incircle Cevians of a triangle

FORMULA

48x^4 + 16x^3 - 27x^2 - 18x - 3 has real positive root x = 1/4(3^(1/3)+3^(2/3)) = 0.880583348...

16*x^3 - 9*x - 3 is the irreducible polynomial. - Michael Somos, Jun 09 2014

MATHEMATICA

N[Select[x/.Solve[48x^4+16x^3-27x^2-18x-3==0, {x}], Im[#]==0&&Re[#]>0 &], 100]

PROG

(PARI) polrootsreal(16*x^3-9*x-3)[1] \\ Charles R Greathouse IV, Aug 26 2017

CROSSREFS

Sequence in context: A309826 A321096 A345746 * A144802 A275477 A019960

Adjacent sequences:  A243505 A243506 A243507 * A243509 A243510 A243511

KEYWORD

nonn,cons

AUTHOR

Frank M Jackson, Jun 05 2014

STATUS

approved

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Last modified July 25 15:18 EDT 2021. Contains 346290 sequences. (Running on oeis4.)