cp's OEIS Frontend

A188615 Decimal expansion of Brocard angle of side-silver right triangle.

%I A188615 #47 Jun 03 2025 01:02:29
%S A188615 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,
%T A188615 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,
%U A188615 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
%N A188615 Decimal expansion of Brocard angle of side-silver right triangle.
%C A188615 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.)
%C A188615 Archimedes's-like 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
%C A188615 This angle is also the half-angle at the summit of the Kelvin wake pattern traced by a boat. - _Robert FERREOL_, Sep 27 2019
%H A188615 G. C. Greubel, <a href="/A188615/b188615.txt">Table of n, a(n) for n = 0..5000</a>
%H A188615 Clark Kimberling, <a href="http://www.heldermann.de/JGG/JGG11/JGG112/jgg11014.htm">Two kinds of golden triangles, generalized to match continued fractions</a>, Journal for Geometry and Graphics, 11 (2007) 165-171.
%H A188615 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wake#Kelvin_wake_pattern">Kelvin wake pattern</a>
%H A188615 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A188615 (Brocard angle) = arccot((a^2+b^2+c^2)/(4*area(ABC))) = arccot(sqrt(8)).
%F A188615 Also equals arcsin(1/3) or arccsc(3). - _Jean-François Alcover_, May 29 2013
%F A188615 Equals Integral_{x=sqrt(2)/2..sqrt(2)} dx/(x^2 + 1). - _Kritsada Moomuang_, May 29 2025
%e A188615 Brocard angle: 0.3398369094541219370963925133917640663882 approx.
%e A188615 Brocard angle: 19.471220634490691369245999 degrees, approx.
%t A188615 r=1+2^(1/2);
%t A188615 b=1; a=r*b; c=(a^2+b^2)^(1/2);
%t A188615 area=(1/4)((a+b+c)(b+c-a)(c+a-b)(a+b-c))^(1/2);
%t A188615 brocard=ArcCot[(a^2+b^2+c^2)/(4area)];
%t A188615 N[brocard, 130]
%t A188615 RealDigits[N[brocard,130]][[1]]
%t A188615 N[180 brocard/Pi,130] (* degrees *)
%t A188615 RealDigits[ArcCos[Sqrt[8/9]], 10, 50][[1]] (* _G. C. Greubel_, Nov 18 2017 *)
%o A188615 (PARI) acos(sqrt(8/9)) \\ _Charles R Greathouse IV_, May 02 2013
%o A188615 (Magma) [Arccos(Sqrt(8/9))]; // _G. C. Greubel_, Nov 18 2017
%Y A188615 Cf. A188614, A188543, A152149.
%K A188615 nonn,cons
%O A188615 0,1
%A A188615 _Clark Kimberling_, Apr 05 2011