login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A188595 Decimal expansion of Brocard angle of side-golden right triangle. 6
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.)

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

FORMULA

Brocard angle: arccot((a^2+b^2+c^2)/(4*area(ABC))) = arccot(sqrt(5)).

EXAMPLE

Brocard angle: 0.420534335283965127888262515913215373 approx.

MATHEMATICA

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]]

RealDigits[ArcTan[Sqrt[1/5]], 10, 50][[1]] (* G. C. Greubel, Nov 21 2017 *)

PROG

(PARI) atan(sqrt(1/5)) \\ G. C. Greubel, Nov 21 2017

(MAGMA) [Arctan(Sqrt(1/5))]; // G. C. Greubel, Nov 21 2017

CROSSREFS

Cf. A188594, A152149, A188615.

Sequence in context: A077116 A249507 A135730 * A144102 A195388 A255324

Adjacent sequences:  A188592 A188593 A188594 * A188596 A188597 A188598

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Apr 05 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 17 22:26 EDT 2019. Contains 328134 sequences. (Running on oeis4.)