A188595 -id:A188595 - cp's OEIS frontend

A188594 Decimal expansion of (circumradius)/(inradius) of side-golden right triangle.

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

Offset: 1

Clark Kimberling, Apr 05 2011

This ratio 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 collection of squares.)

Largest root of 4*x^4 - 20*x^2 - 20*x - 5. - Charles R Greathouse IV, May 07 2011

			2.656875757337521549489732...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Cf. A001622, A152149, A188595, A188614.

phi := (1+Sqrt(5))/2; [(Sqrt(5) + phi*Sqrt(2 + phi))/2]; // G. C. Greubel, Nov 23 2017

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);
RealDigits[N[a*b*c*(a+b+c)/(8*area^2), 130]][[1]]
RealDigits[(Sqrt[5] + GoldenRatio*Sqrt[2 + GoldenRatio])/(2),10,50][[1]] (* G. C. Greubel, Nov 23 2017 *)

{phi = (1 + sqrt(5))/2}; (sqrt(5) + phi*sqrt(2 + phi))/2 \\ G. C. Greubel, Nov 23 2017

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

Equals (sqrt(5) + phi*sqrt(2 + phi))/2, where phi = A001622 is the golden ratio. - G. C. Greubel, Nov 23 2017

A195720 Decimal expansion of arccos(sqrt(1/6)) and of arcsin(sqrt(5/6)) and arctan(sqrt(5)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arccos(sqrt(1/6)) = 1.150261991510931...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Kunle Adegoke, Infinite arctangent sums involving Fibonacci and Lucas numbers, arXiv:1603.08097 [math.NT], 2016.
Index entries for transcendental numbers

Cf. A188595, A195721, A195722.

[Arccos(Sqrt(1/6))]; // G. C. Greubel, Nov 23 2017

r = Sqrt[1/6]; RealDigits[ArcCos[r], 10, 100][[1]]

atan(sqrt(5)) \\ Michel Marcus, Mar 29 2016

Equals Sum_{k >= 1} sqrt(5)/L(2n) where L=A000032. See also A005248. - Michel Marcus, Mar 29 2016

A195708 Decimal expansion of arccos(sqrt(2/5)) and of arcsin(sqrt(3/5)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

			0.886077123792...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A073000, A105199, A195701.

[Arccos(Sqrt(2/5))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/5]; s = Sqrt[2/5];
N[ArcSin[r], 100]
RealDigits[%]  (* A073000 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A105199 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188595 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A137218 *)
N[ArcSin[s], 100]
RealDigits[%]  (* A195701 *)
N[ArcCos[s], 100]
RealDigits[%]  (* A195708 *)
N[ArcTan[s], 100]
RealDigits[%]  (* A195709 *)
N[ArcCos[-s], 100]
RealDigits[%]  (* A195710 *)

acos(sqrt(2/5)) \\ G. C. Greubel, Nov 18 2017

Equals arctan(sqrt(3/2)). - Amiram Eldar, Jul 04 2023

A195709 Decimal expansion of arctan(sqrt(2/5)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

			arctan(sqrt(2/5)) = 0.5639426413606...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A195708.

[Arctan(Sqrt(2/5))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/5]; s = Sqrt[2/5];
N[ArcSin[r], 100]
RealDigits[%]  (* A073000 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A105199 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188595 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A137218 *)
N[ArcSin[s], 100]
RealDigits[%]  (* A195701 *)
N[ArcCos[s], 100]
RealDigits[%]  (* A195708 *)
N[ArcTan[s], 100]
RealDigits[%]  (* A195709 *)
N[ArcCos[-s], 100]
RealDigits[%]  (* A195710 *)

atan(sqrt(2/5)) \\ G. C. Greubel, Nov 18 2017

A195710 Decimal expansion of arccos(-sqrt(2/5)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arccos(-sqrt(2/5)) = 2.25551552979717...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A195708, A195701.

[Arccos(-Sqrt(2/5))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/5]; s = Sqrt[2/5];
N[ArcSin[r], 100]
RealDigits[%]  (* A073000 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A105199 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188595 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A137218 *)
N[ArcSin[s], 100]
RealDigits[%]  (* A195701 *)
N[ArcCos[s], 100]
RealDigits[%]  (* A195708 *)
N[ArcTan[s], 100]
RealDigits[%]  (* A195709 *)
N[ArcCos[-s], 100]
RealDigits[%]  (* A195710 *)
RealDigits[ArcCos[-Sqrt[(2/5)]],10,120][[1]] (* Harvey P. Dale, Apr 06 2023 *)

acos(-sqrt(2/5)) \\ G. C. Greubel, Nov 18 2017

Equals Pi - arcsin(sqrt(3/5)) = Pi - arctan(sqrt(3/2)). - Amiram Eldar, Jul 08 2023

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A188594 Decimal expansion of (circumradius)/(inradius) of side-golden right triangle.

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A195720 Decimal expansion of arccos(sqrt(1/6)) and of arcsin(sqrt(5/6)) and arctan(sqrt(5)).

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A195708 Decimal expansion of arccos(sqrt(2/5)) and of arcsin(sqrt(3/5)).

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Magma

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A195709 Decimal expansion of arctan(sqrt(2/5)).

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Magma

Mathematica

PARI

A195710 Decimal expansion of arccos(-sqrt(2/5)).

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Magma

Mathematica

PARI

Formula