A139349 -id:A139349 - cp's OEIS frontend

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

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

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

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

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

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