A139342 -id:A139342 - cp's OEIS frontend

A139341 Decimal expansion of e^((1+sqrt(5))/2).

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

Offset: 1

Mohammad K. Azarian, Apr 14 2008

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

			5.04316564336002865131188218928542471032359017541384...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139342.

phi := (1+sqrt(5))/2 ; evalf(exp(phi)) ; # R. J. Mathar, Oct 16 2015

RealDigits[Exp[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Feb 08 2022 *)

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

From Amiram Eldar, Feb 08 2022: (Start)
Equals exp(A001622).
Equals 1/A139342. (End)

A139345 Decimal expansion of sine of the golden ratio. That is, the decimal expansion of sin((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

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

			0.99888450909488479883326824263012904463865119212705...

Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3 (Fall 1998), p. 176; Solution, ibid., Vol. 12, No. 1 (Winter 2000), pp. 61-62.
Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139350.

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

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

Equals sin(A001622).

Equals 1/A139350. - Amiram Eldar, Feb 07 2022

Leading zero removed by R. J. Mathar, Feb 05 2009

A139346 Decimal expansion of cosine of the golden ratio, negated. That is, the decimal expansion of -cos((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

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

			-0.04722009625435983376687869404879456549554899472734...

Index entries for transcendental numbers.

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

Join[{0}, RealDigits[Cos[GoldenRatio], 10, 100][[1]]] (* Amiram Eldar, Feb 07 2022 *)

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

Equals 1/A139349. - Amiram Eldar, Feb 07 2022

Edited by N. J. A. Sloane, Dec 11 2008

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

Original entry on oeis.org

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

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

A328344 Decimal expansion of Sum_{k>=0} L(k)/k!, where L(k) is the k-th Lucas number (A000032).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 22 2019

			5.582168726084073272273820087650101080679708682257948...

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd edition, Wiley, 2017, chapter 13.8, pp. 248-250.

Cf. A000032, A001113, A001622, A098689, A139341, A139342, A328495.

RealDigits[Exp[GoldenRatio] + Exp[1 - GoldenRatio], 10, 100][[1]]

Equals exp(phi) + exp(1-phi), where phi is the golden ratio (A001622).

Equals e * A328495. - Amiram Eldar, Feb 06 2022

A328495 Decimal expansion of Sum_{k>=0} (-1)^k*L(k)/k!, where L(k) is the k-th Lucas number (A000032).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 22 2019

			2.053565111476510960344914661146965309320258644945918...

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd edition, Wiley, 2017, chapter 13.8, pp. 248-250.

Cf. A000032, A001113, A001622, A099935, A139341, A139342, A328344, A249455.

Digits := 100: 2*exp(-1/2)*cosh(sqrt(5)/2)*10^86:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Oct 22 2019

RealDigits[Exp[-GoldenRatio] + Exp[GoldenRatio - 1], 10, 100][[1]]

Equals exp(-phi) + exp(phi-1), where phi is the golden ratio (A001622).
Equals 2*exp(-1/2)*cosh(sqrt(5)/2) = A249455*cosh(phi - 1/2). - Peter Luschny, Oct 22 2019
Equals A328344 / e. - Amiram Eldar, Feb 06 2022

cp's OEIS Frontend

A139341 Decimal expansion of e^((1+sqrt(5))/2).

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A139345 Decimal expansion of sine of the golden ratio. That is, the decimal expansion of sin((1+sqrt(5))/2).

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A139346 Decimal expansion of cosine of the golden ratio, negated. That is, the decimal expansion of -cos((1+sqrt(5))/2).

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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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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.