This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A188595 #26 Feb 11 2025 12:23:21 %S A188595 4,2,0,5,3,4,3,3,5,2,8,3,9,6,5,1,2,7,8,8,8,2,6,2,5,1,5,9,1,3,2,1,5,3, %T A188595 7,3,3,5,1,0,3,9,3,9,2,8,1,9,9,1,9,6,0,9,8,8,9,2,6,1,4,0,2,3,4,6,0,4, %U A188595 4,6,5,1,7,3,8,1,6,8,6,8,0,2,5,9,2,6,7,0,0,2,4,2,5,7,9,2,5,1,6,8,9,1,4,8,9,3,4,2,6,1,8,0,1,5,2,5,8,0,2,5,2,1,1,7,7,8,2,0,6,8 %N A188595 Decimal expansion of Brocard angle of side-golden right triangle. %C A188595 The Brocard angle 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 nest of squares.) %C A188595 Also <3_5> in Conway et al. (1999). - _Eric W. Weisstein_, Nov 06 2024 %H A188595 G. C. Greubel, <a href="/A188595/b188595.txt">Table of n, a(n) for n = 0..5000</a> %H A188595 John H. Conway, Charles Radin, and Lorenzo Sadun, <a href="https://arxiv.org/abs/ph/9812019">On Angles Whose Squared Trigonometric Functions are Rational</a>, arXiv:math-ph/9812019, 1998. See also <a href="https://doi.org/10.1007/PL00009463">Discr. Computat. Geom.</a> (1999) Vol. 22, 321-332. %H A188595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DehnInvariant.html">Dehn Invariant</a>. %H A188595 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>. %F A188595 Brocard angle: arccot((a^2+b^2+c^2)/(4*area(ABC))) = arccot(sqrt(5)). %F A188595 Equals A228496/2. - _Hugo Pfoertner_, Nov 06 2024 %e A188595 Brocard angle: 0.420534335283965127888262515913215373 approx. %t A188595 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); brocard = ArcCot[(a^2+b^2+c^2)/(4 area)]; RealDigits[N[brocard,130]][[1]] %t A188595 RealDigits[ArcTan[Sqrt[1/5]], 10, 50][[1]] (* _G. C. Greubel_, Nov 21 2017 *) %o A188595 (PARI) atan(sqrt(1/5)) \\ _G. C. Greubel_, Nov 21 2017 %o A188595 (Magma) [Arctan(Sqrt(1/5))]; // _G. C. Greubel_, Nov 21 2017 %Y A188595 Cf. A188594, A152149, A188615, A228496. %K A188595 nonn,cons %O A188595 0,1 %A A188595 _Clark Kimberling_, Apr 05 2011