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 A188615 Decimal expansion of Brocard angle of side-silver right triangle. 3
 3, 3, 9, 8, 3, 6, 9, 0, 9, 4, 5, 4, 1, 2, 1, 9, 3, 7, 0, 9, 6, 3, 9, 2, 5, 1, 3, 3, 9, 1, 7, 6, 4, 0, 6, 6, 3, 8, 8, 2, 4, 4, 6, 9, 0, 3, 3, 2, 4, 5, 8, 0, 7, 1, 4, 3, 1, 9, 2, 3, 9, 6, 2, 4, 8, 9, 9, 1, 5, 8, 8, 8, 6, 6, 4, 8, 4, 8, 4, 1, 1, 4, 6, 0, 7, 6, 5, 7, 9, 2, 5, 0, 0, 1, 9, 7, 6, 1, 2, 8, 5, 2, 1, 2, 9, 7, 6, 3, 8, 0, 7, 4, 0, 2, 2, 9, 4, 4, 7, 4, 1, 5, 2, 3, 9, 3, 5, 7, 5, 7 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The Brocard angle 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.) REFERENCES Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions," Journal for Geometry and Graphics, 11 (2007) 165-171. LINKS FORMULA (Brocard angle)=arccot((a^2+b^2+c^2)/(4*area(ABC)))=arccot(sqrt(8)). EXAMPLE Brocard angle: 0.3398369094541219370963925133917640663882 approx. Brocard angle: 19.471220634490691369245999 degrees, approx. 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); brocard=ArcCot[(a^2+b^2+c^2)/(4area)]; N[brocard, 130] RealDigits[N[brocard, 130]][[1]] N[180 brocard/Pi, 130] (* degrees *) PROG (PARI) acos(sqrt(8/9)) \\ Charles R Greathouse IV, May 02 2013 CROSSREFS Cf. A188614, A188543, A152149. Sequence in context: A065483 A019745 A173815 * A155686 A201456 A064235 Adjacent sequences:  A188612 A188613 A188614 * A188616 A188617 A188618 KEYWORD nonn,cons AUTHOR Clark Kimberling, Apr 05 2011 STATUS approved

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Last modified May 21 13:58 EDT 2013. Contains 225489 sequences.