A188543 -id:A188543 - cp's OEIS frontend

A152149 Decimal expansion of the angle B in the doubly golden triangle ABC.

6, 5, 7, 4, 0, 5, 4, 8, 2, 9, 7, 6, 5, 3, 2, 5, 9, 2, 3, 8, 0, 9, 6, 8, 5, 4, 1, 5, 2, 9, 3, 9, 7, 1, 2, 6, 5, 4, 1, 4, 9, 5, 9, 4, 6, 4, 8, 7, 8, 3, 9, 3, 7, 0, 7, 8, 2, 0, 9, 2, 8, 0, 8, 5, 8, 8, 5, 3, 9, 5, 0, 6, 1, 3, 8, 1, 7, 7, 3, 5, 0, 7, 0, 1, 7, 1, 5, 1, 6, 5, 4, 4, 0, 5, 2, 2, 7, 8, 0, 5, 2, 8, 1, 2, 6

Offset: 0

Clark Kimberling, Nov 26 2008

There is a unique (shape of) triangle ABC that is both side-golden and angle-golden. Its angles are B, C=t*B and A=pi-B-t*B, where t is the golden ratio. "Angle-golden" and "side-golden" refer to partitionings of ABC, each in a manner that matches the continued fraction [1,1,1,...] of t.  (The partitionings are analogous to the partitioning of the golden rectangle into squares by the removal of exactly 1 square at each stage.)
For doubly silver and doubly e-ratio triangles, see A188543 and A188544.
For the side partitioning and angle partitioning (i,e, constructions) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.

			The number B begins with 0.65740548 (equivalent to 37.666559... degrees).

Clark Kimberling, "A new kind of golden triangle," in Applications of Fibonacci Numbers, Proc. Fourth International Conference on Fibonacci Numbers and Their Applications, Kluwer, 1991.

Iain Fox, Table of n, a(n) for n = 0..20000
Jordi Dou, Clark Kimberling and Laurence Kuipers, A Fibonacci sequence of nested triangles, Problem S29, Amer. Math. Monthly 89 (1982) 696-697; proposed 87 (1980) 302.
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Cf. A000045, A188543, A188544, A376961 (length of shortest side of the doubly golden triangle that has area 1).

r = (1 + 5^(1/2))/2; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]

t=(1+5^(1/2))/2; solve(b=.6, .7, sin(b*t^2)-t*sin(b)) \\ Iain Fox, Feb 11 2020

B is the number in [0,Pi] such that sin(B*t^2)=t*sin(B), where t=(1+5^(1/2))/2, the golden ratio.

Keyword:cons added and offset corrected by R. J. Mathar, Jun 18 2009

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

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Apr 05 2011

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.)
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
This angle is also the half-angle at the summit of the Kelvin wake pattern traced by a boat. - Robert FERREOL, Sep 27 2019

			Brocard angle: 0.3398369094541219370963925133917640663882 approx.
Brocard angle: 19.471220634490691369245999 degrees, approx.

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) 165-171.
Wikipedia, Kelvin wake pattern
Index entries for transcendental numbers.

Cf. A188614, A188543, A152149.

[Arccos(Sqrt(8/9))]; // G. C. Greubel, Nov 18 2017

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 *)
RealDigits[ArcCos[Sqrt[8/9]], 10, 50][[1]] (* G. C. Greubel, Nov 18 2017 *)

acos(sqrt(8/9)) \\ Charles R Greathouse IV, May 02 2013

(Brocard angle) = arccot((a^2+b^2+c^2)/(4*area(ABC))) = arccot(sqrt(8)).
Also equals arcsin(1/3) or arccsc(3). - Jean-François Alcover, May 29 2013
Equals Integral_{x=sqrt(2)/2..sqrt(2)} dx/(x^2 + 1). - Kritsada Moomuang, May 29 2025

A188614 Decimal expansion of (circumradius)/(inradius) of side-silver right triangle.

Original entry on oeis.org

3, 2, 6, 1, 9, 7, 2, 6, 2, 7, 3, 9, 5, 6, 6, 8, 5, 6, 1, 0, 5, 8, 0, 5, 5, 1, 0, 3, 0, 0, 3, 2, 7, 4, 6, 5, 2, 2, 1, 4, 5, 0, 5, 1, 2, 7, 1, 0, 4, 2, 3, 3, 0, 4, 5, 4, 0, 6, 8, 7, 5, 2, 0, 0, 5, 5, 1, 8, 0, 2, 4, 9, 3, 4, 6, 4, 3, 1, 1, 7, 5, 6, 2, 8, 0, 0, 6, 7, 4, 0, 4, 0, 2, 8, 3, 3, 0, 7, 6, 4, 9, 3, 0, 9, 3, 9, 8, 9, 7, 7, 9, 5, 4, 0, 8, 0, 6, 3, 0, 8, 6, 6, 6, 3, 1, 9, 1, 2, 1, 5

Offset: 1

Clark Kimberling, Apr 05 2011

This ratio 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.)

			ratio=3.26197262739566856105805510300327465221450 approx.

Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Cf. A188543, A152149, A188614, A188618.

a179260 := sqrt(2+sqrt(2)) ; a014176 := 1+sqrt(2) ; 1/(a014176/a179260-1) ; evalf(%) ; # R. J. Mathar, Apr 05 2011

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); RealDigits[N[a*b*c*(a+b+c)/(8*area^2),130]][[1]]

(circumradius)/(inradius) = abc(a+b+c)/(8*area^2), where area=area(ABC).

A188544 Decimal expansion of the angle B in the doubly e-ratio triangle ABC.

Original entry on oeis.org

3, 6, 8, 9, 3, 1, 2, 7, 4, 9, 4, 7, 8, 0, 5, 8, 4, 2, 6, 5, 1, 9, 1, 1, 2, 7, 2, 6, 8, 8, 6, 4, 0, 8, 5, 7, 1, 8, 6, 8, 3, 4, 4, 2, 8, 8, 3, 5, 2, 6, 1, 9, 0, 6, 5, 9, 8, 5, 4, 6, 2, 1, 2, 1, 1, 1, 1, 7, 6, 5, 9, 8, 7, 5, 6, 8, 4, 9, 0, 6, 0, 6, 7, 0, 1, 2, 1, 0, 6, 0, 4, 8, 5, 9, 8, 4, 2, 8, 4, 2, 8, 0, 9, 9, 1, 1, 8, 2, 1, 8, 8, 5, 9, 6, 9, 8, 4, 4, 2, 2, 9, 4, 8, 7, 3, 4, 6, 8

Offset: 0

Clark Kimberling, Apr 03 2011

There is a unique (shape of) triangle ABC that is both side-e-ratio and angle-e-ratio. Its angles are B, t*B and pi-B-t*B, where t=e. "Side-e-ratio" and "angle-e-ratio" refer to partitionings of ABC, each in a manner that matches the continued fraction [2,1,2,1,1,4,1,1,6,...] of t. For doubly golden and doubly silver triangles, see A152149 and A188543. For the side partitioning and angle partitioning (i,e, constructions) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.

			B=0.36893127494780584265191127268864 approximately.
B=21.1382 degrees approximately.

Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Cf. A152149, A188543.

r = E; Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]

B is the number in [0,Pi] such that sin(B*e^2)=e*sin(B).

a(127) corrected by Sean A. Irvine, Sep 08 2021

A188616 Decimal expansion of angle B of unique side-golden and angle-silver triangle.

Original entry on oeis.org

5, 9, 1, 0, 6, 7, 7, 9, 9, 7, 0, 5, 1, 6, 4, 8, 7, 9, 7, 6, 3, 2, 3, 2, 3, 7, 4, 1, 9, 6, 6, 2, 1, 7, 2, 3, 6, 0, 5, 4, 9, 7, 8, 5, 3, 1, 4, 6, 5, 8, 3, 4, 0, 5, 9, 0, 5, 0, 3, 1, 3, 2, 9, 0, 3, 6, 5, 9, 4, 6, 1, 4, 7, 0, 8, 5, 5, 8, 0, 0, 1, 2, 5, 4, 3, 4, 3, 8, 2, 2, 5, 8, 1, 9, 1, 6, 4, 3, 1, 2, 6, 6, 0, 3, 6, 8, 6, 5, 6, 4, 1, 3, 8, 1, 5, 7, 7, 8, 3, 7

Offset: 0

Clark Kimberling, Apr 05 2011

Let r=(golden ratio)=(1+sqrt(5))/2 and u=(silver ratio)=1+sqrt(2). A triangle ABC with sidelengths a,b,c is side-golden if a/b=r and angle-silver if C/B=u. There is a unique triangle that has both properties. The quickest way to understand the geometric reasons for the names is by analogy to the golden and silver rectangles. For the former, exactly 1 square is available at each stage of the partitioning of the rectangle into a nest of squares, and for the former, exactly 2 squares are available. Analogously, for ABC, exactly one 1 triangle of a certain kind is available at each stage of a side-partitioning procedure, and exactly 2 triangles of another kind are available for angle-partitioning. For details, see the 2007 reference.

			B=0.59106779970516487976323237419662 approximately

Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Cf. A152149, A188543.

Remove["Global`*"]; r=1+2^(1/2); u=(1+5^(1/2))/2; RealDigits[FindRoot[Sin[r*t+t]==u*Sin[t],{t,1}, WorkingPrecision->120][[1,2]]][[1]]

A188617 Decimal expansion of angle B of unique side-silver and angle-golden triangle.

Original entry on oeis.org

2, 8, 5, 0, 8, 8, 7, 3, 0, 0, 4, 8, 6, 1, 0, 5, 5, 3, 7, 1, 4, 5, 6, 0, 9, 1, 3, 7, 8, 0, 2, 1, 6, 3, 3, 7, 0, 2, 4, 0, 0, 1, 0, 2, 5, 6, 9, 7, 6, 7, 5, 9, 1, 4, 1, 8, 1, 0, 0, 4, 0, 5, 1, 3, 3, 9, 0, 9, 0, 3, 9, 6, 5, 6, 7, 1, 4, 0, 1, 1, 5, 5, 4, 0, 7, 0, 3, 8, 4, 5, 0, 1, 3, 8, 3, 1, 0, 8, 0, 1, 6, 1, 4, 0, 7, 1, 6, 0, 9, 8, 8, 9, 3, 6, 8, 9, 1, 7, 6, 9

Offset: 0

Clark Kimberling, Apr 05 2011

Let r=(silver ratio)=1+sqrt(2) and u=(golden ratio)=(1+sqrt(5))/2. A triangle ABC with sidelengths a,b,c is side-silver if a/b=r and angle-golden if C/B=u. There is a unique triangle that has both properties. The quickest way to understand the geometric reasons for the names is by analogy to the golden and silver rectangles. For the former, exactly 1 square is available at each stage of the partitioning of the rectangle into a nest of squares, and for the former, exactly 2 squares are available. Analogously, for ABC, exactly one 2 triangles of a certain kind are available at each stage of a side-partitioning procedure, and exactly 1 triangle of another kind are available for angle-partitioning. For details, see the 2007 reference.

			B=0.285088730048610553714560913780216337024001 approximately.

Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Cf. A188616, A152149, A188543.

r=(1+5^(1/2))/2; u=1+2^(1/2); Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == u*Sin[t], {t, 1}, WorkingPrecision->120][[1, 2]]][[1]]

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A152149 Decimal expansion of the angle B in the doubly golden triangle ABC.

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

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A188614 Decimal expansion of (circumradius)/(inradius) of side-silver right triangle.

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A188544 Decimal expansion of the angle B in the doubly e-ratio triangle ABC.

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A188616 Decimal expansion of angle B of unique side-golden and angle-silver triangle.

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A188617 Decimal expansion of angle B of unique side-silver and angle-golden triangle.

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