A094214 -id:A094214 - cp's OEIS frontend

A139347 Decimal expansion of negated tangent of the golden ratio. That is, the decimal expansion of -tan((1+sqrt(5))/2).

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

Offset: 2

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			-21.15380078249327464858628117032582559788124367464826...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139348.

RealDigits[Tan[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Feb 07 2022 *)

-tan((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals tan(A001622).
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A139348.
Equals A139345/A139346. (End)

Offset corrected by Mohammad K. Azarian, Dec 13 2008

Sign added to definition by R. J. Mathar, Feb 05 2009

A139348 Decimal expansion of negated cotangent of the golden ratio. That is, the decimal expansion of -cot((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.04727282866479448118935650960621633420056105722556...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347.

Join[{0},RealDigits[-Cot[GoldenRatio],10,120][[1]]] (* Harvey P. Dale, Sep 18 2013 *)

-cotan((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals cot(A001622).
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A139347.
Equals A139346/A139345. (End)

Added sign in definition. Leading zero dropped by R. J. Mathar, Feb 05 2009

A272535 Decimal expansion of the edge length of a regular 16-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

Like all m-gons with m equal to a power of 2 (see A003401 and A000079), this is a constructible number.

			0.390180644032256535696569736954044481855383235503909615509004...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 8.

Cf. A000079, A003401.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/16], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/16)
```

Equals 2*sin(Pi/m) for m=16, 2*A232738. Equals also sqrt(2-sqrt(2+sqrt(2))).

A272536 Decimal expansion of the edge length of a regular 20-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

Since 20-gon is constructible (see A003401), this is a constructible number.

			0.3128689300804617380202106389343337846277997841713215801692826921...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 8.

Cf. A003401.
Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17).
Cf. A019818.

RealDigits[N[2Sin[Pi/20], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/20)
```

Equals 2*sin(Pi/20) = 2*A019818.
Equals also (sqrt(2)+sqrt(10)-2*sqrt(5-sqrt(5)))/4.
Equals i^(9/10) + i^(-9/10). - Gary W. Adamson, Jul 08 2022

A089959 a(n) = floor(1/(f(n) - f(n)^2)) with f(n) = frac(n*(sqrt(5) - 1)/2) (fractional part).

Original entry on oeis.org

4, 5, 8, 4, 12, 4, 4, 19, 4, 6, 6, 4, 30, 4, 5, 10, 4, 9, 5, 4, 48, 4, 5, 7, 4, 15, 4, 4, 14, 4, 7, 5, 4, 77, 4, 5, 8, 4, 10, 4, 4, 24, 4, 6, 6, 4, 22, 4, 4, 11, 4, 8, 5, 4, 124, 4, 5, 7, 4, 13, 4, 4, 16, 4, 7, 6, 4, 39, 4, 5, 9, 4, 9, 5, 4, 35, 4, 6, 6, 4

Offset: 1

Gary W. Adamson, Nov 16 2003

nonn

Denote by Fn and Ln the Fibonacci resp. Lucas numbers. Then some of the terms follow one of the following two patterns: (1) a(Fn) = (Ln + 1). Example: a(8) = 19 since 8 = F6 and 18 = L6. (2) a(Ln) = (Fn + 1). Example: a(29) = 14 = (F7 + 1) = (13 + 1).

			a(7) = floor(1/({7*k}*(1 - {7*k}))) = floor(1/({4.326...}*(1 - {4.326...}))) = floor(1/(0.326...*0.673...)) = floor(4.549...) = 4.

Cf. A094214.

Table[Floor[1/(FractionalPart[(2*n)/(1+Sqrt[5])]*(1-FractionalPart[ (2*n)/(1 + Sqrt[5])]))], {n, 1, 80}] (* Stefan Steinerberger, Jul 01 2007 *)

default(realprecision,200);p=(sqrt(5)-1)/2;vector(100,n,1\(frac(n*p)-frac(n*p)^2)) \\ M. F. Hasler, Apr 06 2009

a(n) = floor(1/({n*k}*(1 - {n*k}))); k = (sqrt(5) - 1)/2; where {x} = fractional part of x.

Corrected and extended by Stefan Steinerberger, Jul 01 2007

Definition, comment and example reworded and corrected by M. F. Hasler, Apr 06 2009

A139349 Decimal expansion of negated secant of the golden ratio. That is, the decimal expansion of -sec((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 2

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			21.17742400636614440872804040937130213307185355364174...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347, A139348.

RealDigits[-Sec[GoldenRatio],10,120][[1]] (* Harvey P. Dale, Dec 08 2011 *)

-1/cos((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals sec(A001622).

Equals 1/A139346. - Amiram Eldar, Feb 07 2022

Offset corrected by Mohammad K. Azarian, Dec 13 2008

Sign in definition added by R. J. Mathar, Feb 05 2009

A139350 Decimal expansion of csc((1+sqrt(5))/2), where (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			1.00111673661465225489616711351705587794461531806624...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347, A139348, A139349.

RealDigits[Csc[GoldenRatio], 10, 100][[1]] (* Vincenzo Librandi, Feb 19 2013 *)

1/sin((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals 1/A139345. - Amiram Eldar, Feb 07 2022

Edited by Bruno Berselli, Feb 19 2013

A140232 a(n) = ceiling(n*exp((1+sqrt(5))/2)).

Original entry on oeis.org

6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, 91, 96, 101, 106, 111, 116, 122, 127, 132, 137, 142, 147, 152, 157, 162, 167, 172, 177, 182, 187, 192, 197, 202, 207, 212, 217, 222, 227, 232, 238, 243, 248, 253, 258, 263, 268, 273, 278, 283

Offset: 1

Mohammad K. Azarian, May 13 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347, A139348, A139349, A139350, A140231, A016861.

phi:=(1+Sqrt(5))/2; [Ceiling(n*Exp(phi)): n in [1..60]]; // G. C. Greubel, Jun 30 2019

Ceiling[Exp[GoldenRatio]*Range[60]] (* G. C. Greubel, Jun 30 2019 *)

phi=(1+sqrt(5))/2; vector(60, n, ceil(n*exp(phi)) ) \\ G. C. Greubel, Jun 30 2019

[ceil(n*exp(golden_ratio)) for n in (1..60)] # G. C. Greubel, Jun 30 2019

a(n) = ceiling(n*A139341). - R. J. Mathar, Feb 06 2009

A152115 Decimal expansion of the dilogarithm of (the golden mean minus 1), Li_2(phi-1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Nov 24 2008

Equals Li_2(phic) = L(phic)-log(phic)*log(1-phic)/2 = A002388/10 - A002390^2, where Li_2(.) is the dilogarithm, L(.) is Roger's dilogarithm, where phic = phi-1 = A094214, where -log(phic)= A002390 = log(1-phic)/2.

			Equals 0.7553956195317414693865200287560823535149635906747...

L. B. W. Jolley, Summation of Series, Dover (1961)

Anatol N. Kirillov, Dilogarithm identities, arXiv:hep-th/9408113.
J. H. Loxton, Special values of the dilogarithm function, Acta Arithm. 43 (2) (1984), 155-166.

RealDigits[ PolyLog[2, (Sqrt[5]-1)/2], 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

PARI
```
phic=(sqrt(5)-1)/2 ; dilog(phic);
```

Equals sum_{n>=1} x^n/n^2 for x= 2*sin(Pi/10). [Jolley eq (360d)]

More terms from Jean-François Alcover, Feb 12 2013

A189378 a(n) = n + [nr/s] + [nt/s]; r=2, s=(-1+sqrt(5))/2, t=(1+sqrt(5))/2.

Original entry on oeis.org

6, 13, 19, 26, 34, 40, 47, 53, 61, 68, 74, 81, 89, 95, 102, 108, 116, 123, 129, 136, 142, 150, 157, 163, 170, 178, 184, 191, 197, 205, 212, 218, 225, 233, 239, 246, 252, 259, 267, 273, 280, 286, 294, 301, 307, 314, 322, 328, 335, 341, 349, 356, 362, 369, 375, 383, 390, 396, 403, 411, 417, 424, 430, 438, 445, 451, 458, 466, 472, 479, 485, 492, 500, 506, 513, 519, 527, 534, 540, 547, 555, 561

Offset: 1

Clark Kimberling, Apr 20 2011

nonn

Theorem: These are the numbers k such that (k+1)-sections of the Fibonacci word contain neither "000" nor "111". Proved by J. Shallit and "Walnut", Apr 06 2021. - Don Reble, Apr 06 2021

Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 11-12.

Cf. A189377, A189379.
Cf. A001622, A094214.
See also A339950.

Mathematica
```
(See A189377.)
```

cp's OEIS Frontend

A139347 Decimal expansion of negated tangent of the golden ratio. That is, the decimal expansion of -tan((1+sqrt(5))/2).

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A139348 Decimal expansion of negated cotangent of the golden ratio. That is, the decimal expansion of -cot((1+sqrt(5))/2).

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A272535 Decimal expansion of the edge length of a regular 16-gon with unit circumradius.

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A272536 Decimal expansion of the edge length of a regular 20-gon with unit circumradius.

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A089959 a(n) = floor(1/(f(n) - f(n)^2)) with f(n) = frac(n*(sqrt(5) - 1)/2) (fractional part).

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A139349 Decimal expansion of negated secant of the golden ratio. That is, the decimal expansion of -sec((1+sqrt(5))/2).

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A139350 Decimal expansion of csc((1+sqrt(5))/2), where (1+sqrt(5))/2 is the golden ratio.

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