A188614 Decimal expansion of (circumradius)/(inradius) of side-silver right triangle.

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

Offset: 1

Author

Clark Kimberling, Apr 05 2011

Keywords

• nonn
• cons

Comments

This ratio is invariant of the size of the side-silver 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=(silver ratio)=(1+sqrt(2)). This is the unique right triangle matching the continued fraction [2,2,2,...] of r; i.e., under the side-partitioning procedure described in the 2007 reference, there are exactly 2 removable subtriangles at each stage. (This is analogous to the removal of 2 squares at each stage of the partitioning of the silver rectangle as a nest of squares.)

Examples

			ratio=3.26197262739566856105805510300327465221450 approx.
		

Links

• Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Crossrefs

Cf. A188543, A152149, A188614, A188618.

Programs

Maple

a179260 := sqrt(2+sqrt(2)) ; a014176 := 1+sqrt(2) ; 1/(a014176/a179260-1) ; evalf(%) ; # R. J. Mathar, Apr 05 2011

Mathematica

r= 1+2^(1/2); 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]]

Formula

(circumradius)/(inradius) = abc(a+b+c)/(8*area^2), where area=area(ABC).