

A188594


Decimal expansion of (circumradius)/(inradius) of sidegolden right triangle.


4



2, 6, 5, 6, 8, 7, 5, 7, 5, 7, 3, 3, 7, 5, 2, 1, 5, 4, 9, 4, 8, 9, 7, 3, 2, 1, 2, 2, 3, 8, 4, 0, 9, 3, 0, 2, 9, 7, 2, 3, 6, 6, 0, 2, 5, 1, 5, 7, 4, 6, 5, 9, 0, 7, 5, 6, 5, 5, 0, 2, 6, 7, 4, 7, 8, 9, 2, 6, 9, 2, 1, 0, 7, 0, 6, 6, 4, 4, 7, 9, 0, 8, 9, 3, 4, 5, 0, 4, 0, 6, 5, 0, 2, 2, 9, 4, 3, 8, 5, 5, 1, 2, 0, 7, 0, 6, 9, 3, 7, 2, 2, 9, 5, 4, 2, 5, 5, 5, 3, 2, 7, 4, 5, 2, 6, 3, 0, 3, 8, 1
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OFFSET

1,1


COMMENTS

This ratio is invariant of the size of the sidegolden right triangle ABC. The shape of ABC is given by sidelengths a,b,c, where a=r*b, and c=sqrt(a^2+b^2), where r=(golden ratio)=(1+sqrt(5))/2. This is the unique right triangle matching the continued fraction [1,1,1,...] of r; i.e, under the sidepartitioning procedure described in the 2007 reference, there is exactly 1 removable subtriangle at each stage. (This is analogous to the removal of 1 square at each stage of the partitioning of the golden rectangle as a collection of squares.)
Largest root of 4*x^4  20*x^2  20*x  5.  Charles R Greathouse IV, May 07, 2011


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165171.


FORMULA

(circumradius)/(inradius)=abc(a+b+c)/(8*area^2), where area=area(ABC).
Equals (sqrt(5) + phi*sqrt(2 + phi))/2, where phi = A001622 is the golden ratio.  G. C. Greubel, Nov 23 2017


EXAMPLE

2.656875757337521549489732...


MATHEMATICA

r=(1+5^(1/2))/2; b=1; a=r*b; c=(a^2+b^2)^(1/2);
area = (1/4)((a+b+c)(b+ca)(c+ab)(a+bc))^(1/2);
RealDigits[N[a*b*c*(a+b+c)/(8*area^2), 130]][[1]]
RealDigits[(Sqrt[5] + GoldenRatio*Sqrt[2 + GoldenRatio])/(2), 10, 50][[1]] (* G. C. Greubel, Nov 23 2017 *)


PROG

(PARI) {phi = (1 + sqrt(5))/2}; (sqrt(5) + phi*sqrt(2 + phi))/2 \\ G. C. Greubel, Nov 23 2017
(MAGMA) phi := (1+Sqrt(5))/2; [(Sqrt(5) + phi*Sqrt(2 + phi))/2]; // G. C. Greubel, Nov 23 2017


CROSSREFS

Cf. A001622, A152149, A188595, A188614.
Sequence in context: A019655 A032582 A050928 * A107822 A094514 A171031
Adjacent sequences: A188591 A188592 A188593 * A188595 A188596 A188597


KEYWORD

nonn,easy,cons


AUTHOR

Clark Kimberling, Apr 05 2011


STATUS

approved



