

A188615


Decimal expansion of Brocard angle of sidesilver right triangle.


6



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, 6
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OFFSET

0,1


COMMENTS

The Brocard angle is invariant of the size of the sidesilver 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 sidepartitioning 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.)
Archimedes'slike scheme: set p(0) = 1/(2*sqrt(2)), q(0) = 1/3; p(n+1) = 2*p(n)*q(n)/(p(n)+q(n)) (harmonic mean, i.e., 1/p(n+1) = (1/p(n) + 1/q(n))/2), q(n+1) = sqrt(p(n+1)*q(n)) (geometric mean, i.e., log(q(n+1)) = (log(p(n+1)) + log(q(n)))/2), for n >= 0. The error of p(n) and q(n) decreases by a factor of approximately 4 each iteration, i.e., approximately 2 bits are gained by each iteration. Set r(n) = (2*q(n) + p(n))/3, the error decreases by a factor of approximately 16 for each iteration, i.e., approximately 4 bits are gained by each iteration. For a similar scheme see also A244644.  A.H.M. Smeets, Jul 12 2018
This angle is also the halfangle at the summit of the Kelvin wake pattern traced by a boat.  Robert FERREOL, Sep 27 2019


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165171.
Wikipedia, Kelvin wake pattern


FORMULA

(Brocard angle) = arccot((a^2+b^2+c^2)/(4*area(ABC))) = arccot(sqrt(8)).
Also equals arcsin(1/3) or arccsc(3).  JeanFrançois Alcover, May 29 2013


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+ca)(c+ab)(a+bc))^(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 *)
RealDigits[ArcCos[Sqrt[8/9]], 10, 50][[1]] (* G. C. Greubel, Nov 18 2017 *)


PROG

(PARI) acos(sqrt(8/9)) \\ Charles R Greathouse IV, May 02 2013
(MAGMA) [Arccos(Sqrt(8/9))]; // G. C. Greubel, Nov 18 2017


CROSSREFS

Cf. A188614, A188543, A152149.
Sequence in context: A065483 A019745 A173815 * A328566 A155686 A290300
Adjacent sequences: A188612 A188613 A188614 * A188616 A188617 A188618


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Apr 05 2011


STATUS

approved



