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A188594 Decimal expansion of (circumradius)/(inradius) of side-golden 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This ratio is invariant of the size of the side-golden 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 side-partitioning 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) 165-171.

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+c-a)(c+a-b)(a+b-c))^(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

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Last modified December 12 20:04 EST 2017. Contains 295954 sequences.