A188594 Decimal expansion of (circumradius)/(inradius) of side-golden right triangle.

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

Offset: 1

Author

Clark Kimberling, Apr 05 2011

Keywords

• nonn
• easy
• cons

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

Examples

			2.656875757337521549489732...
		

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.

Crossrefs

Cf. A001622, A152149, A188595, A188614.

Programs

Magma

phi := (1+Sqrt(5))/2; [(Sqrt(5) + phi*Sqrt(2 + phi))/2]; // G. C. Greubel, Nov 23 2017

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 *)

PARI

{phi = (1 + sqrt(5))/2}; (sqrt(5) + phi*sqrt(2 + phi))/2 \\ G. C. Greubel, Nov 23 2017

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