A175288 -id:A175288 - cp's OEIS frontend

A195693 Decimal expansion of arctan(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

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

Offset: 0

Clark Kimberling, Sep 22 2011

Radian measure of half the smaller angle in the golden rhombus. - Eric W. Weisstein, Dec 11 2018

The angle between the diagonal and the longer side of a golden rectangle. - Amiram Eldar, May 18 2021

			arctan(1/phi) = 0.5535743588970452515085327300892685200... .
tan(0.5535743588970452515085327300...) = 1/(golden ratio).
cot(0.5535743588970452515085327300...) = (golden ratio).

Paul S. Bruckman, Problem H-549, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 37, No. 1 (1999), p. 91; Resurrection, Solution to Problem H-549 by Charles K. Cook, ibid., Vol. 38, No. 2 (2000), pp. 191-192.
Hei-Chi Chan, Machin-type formulas expressing Pi in terms of phi, The Fibonacci Quarterly, Vol. 46/47, No. 1 (2008/2009), pp. 32-37.
Verner E. Hoggatt, Jr. and I. D. Bruggles, A Primer on the Fibonacci Sequence, Part V, The Fibonacci Quarterly, Vol. 2, No. 1 (1964), pp. 59-65.
Eric Weisstein's World of Mathematics, Golden Rhombus.
Index entries for transcendental numbers

Cf. A000032, A000045, A001622, A005248, A175288, A195692, A195694, A195723.

(See also A195692.)
RealDigits[ArcCot[GoldenRatio], 10, 100][[1]] (* or *) RealDigits[(Pi - ArcTan[4/3])/4, 10, 100][[1]] (* Eric W. Weisstein, Dec 11 2018 *)

PARI
```
atan(2)/2 \\ Michel Marcus, Feb 05 2022
```

Equals Pi/2 - A195723. - Amiram Eldar, May 18 2021
Equals arctan(2)/2. - Christoph B. Kassir, Dec 04 2021
From Amiram Eldar, Jan 11 2022: (Start)
Equals arccot(phi).
Equals (Pi - arctan(phi^5))/3.
Equals (Pi - arctan(4/3))/4.
Equals Sum_{k>=1} ((-1)^(k+1) * arctan(1/Fibonacci(2*k))) (Bruckman, 1999). (End)
Equals Sum_{k>=1} arctan(1/Lucas(2*k)) (Hoggatt and Bruggles, 1964). - Amiram Eldar, Feb 05 2022

A195692 Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 22 2011

Every cyclic quadrilateral all of whose angles are greater than arccos((sqrt(5)-1)/2) admits a 3 × 1 grid dissection into three cyclic quadrilaterals [Thm. 2.3 in Choi et al. p. 2]. - Michel Marcus, Aug 13 2019
The base angle of the isosceles triangle of smallest perimeter which circumscribes a semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022
Smallest positive root of the equation sin(x) = cot(x). - Wolfe Padawer, Apr 11 2023

			arccos(1/phi) = 0.904556894302381364127316795661958721...
cos(0.904556894302381364127316795661958721...) = 1/(golden ratio) = 0.618...
sec(0.904556894302381364127316795661958721...) = (golden ratio) = 1.618...

Erica Choi, Dan Ismailescu, Jiho Lee and Joonsoo Lee, Grid dissections of tangential quadrilaterals, arXiv:1908.02251 [math.MG], 2019.
Duane W. DeTemple, The Triangle of Smallest Perimeter which Circumscribes a Semicircle, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
Index entries for transcendental numbers

Cf. A001622, A019669, A139339, A175288, A195693, A195694, A197762.

r = 1/GoldenRatio;
N[ArcCos[r], 100]
RealDigits[%]

acos(2/(sqrt(5)+1)) \\ Charles R Greathouse IV, Nov 21 2024

From Amiram Eldar, Feb 07 2022: (Start)
Equals Pi/2 - A175288.
Equals arcsin(1/sqrt(phi)).
Equals arctan(sqrt(phi)). (End)

Terms replaced with intended terms by Rick L. Shepherd, Jan 30 2013

A195694 Decimal expansion of arccos(-1/r), where r = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 22 2011

Let c1 and c2 be the constants at A195692 and A195694; then c1+c2 = Pi.

			arccos(-1/r) = 2.2370357592874118743353265876175441...

Index entries for transcendental numbers

Cf. A001622, A175288, A195692, A195693.

(See A195692.)
RealDigits[ArcCos[-1/GoldenRatio],10,120][[1]] (* Harvey P. Dale, Jan 15 2012 *)

acos(-2/(sqrt(5)+1)) \\ Charles R Greathouse IV, Nov 21 2024

A352151 Expansion of e.g.f. 1/(cos(x) - tan(x)).

Original entry on oeis.org

1, 1, 3, 14, 81, 616, 5523, 58064, 697281, 9417856, 141368643, 2334020864, 42039523281, 820296426496, 17237259945363, 388087200241664, 9320064293358081, 237814050877505536, 6425096888209255683, 183232685725482942464, 5500505587921088841681

Offset: 0

Seiichi Manyama, Mar 06 2022

nonn

Cf. A000182, A000828, A001586, A175288, A352164.

m = 20; Range[0, m]! * CoefficientList[Series[1/(Cos[x] - Tan[x]), {x, 0, m}], x] (* Amiram Eldar, Mar 06 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(cos(x)-tan(x))))

c(n) = ((-4)^n-(-16)^n)*bernfrac(2*n)/(2*n);
b(n) = if(n%2==1, c((n+1)/2), (-1)^(n/2+1));
a(n) = if(n==0, 1, sum(k=1, n, b(k)*binomial(n, k)*a(n-k)));

a(0) = 1; a(n) = Sum_{k=1..n} b(k) * binomial(n,k) * a(n-k), where b(k) = A000182((k+1)/2) if k is odd, otherwise (-1)^(k/2+1).
From Vaclav Kotesovec, Mar 06 2022: (Start)
a(n) ~ n! / (sqrt(5) * (arctan(sqrt((sqrt(5) - 1)/2)))^(n+1)).
a(n) ~ n! / (sqrt(5) * A175288^(n+1)). (End)

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A195693 Decimal expansion of arctan(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

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A195692 Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

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A195694 Decimal expansion of arccos(-1/r), where r = (1+sqrt(5))/2 (the golden ratio).

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A352151 Expansion of e.g.f. 1/(cos(x) - tan(x)).

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