A073448 -id:A073448 - cp's OEIS frontend

A049470 Decimal expansion of cos(1).

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

Offset: 0

Albert du Toit (dutwa(AT)intekom.co.za), N. J. A. Sloane

Also, decimal expansion of the real part of e^i. - Bruno Berselli, Feb 08 2013

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

			0.5403023058681397...

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, p. 413-414.
Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 10 (formula 0.245.7).
Simon Plouffe, cos(1)
Eric Weisstein's World of Mathematics, Factorial Sums
Index entries for transcendental numbers

Cf. A049469 (imaginary part of e^i), A211883 (real part of -(i^e)), A211884 (imaginary part of -(i^e)). - Bruno Berselli, Feb 08 2013
Cf. A073743 ( cosh(1) ), A073448, A275651.
Cf. A068985, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

evalf(cos(1)); # Altug Alkan, Sep 22 2018

Mathematica
```
RealDigits[Cos[1], 10, 110] [[1]]
```

cos(1) \\ Charles R Greathouse IV, Jan 04 2016

Continued fraction representation: cos(1) = 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/((4*n^2 - 2*n - 1) + ... ))))). See A275651 for proof. Cf. A073743. - Peter Bala, Sep 02 2016
Equals Sum_{k >= 0} (-1)^k/A010050(k), where A010050(k) = (2k)! [See Gradshteyn and Ryzhik]. - A.H.M. Smeets, Sep 22 2018
Equals 1/A073448. - Alois P. Heinz, Jan 23 2023
From Gerry Martens, May 04 2024: (Start)
Equals (4*(cos(1/4)^4 + sin(1/4)^4) - 3).
Equals (16*(cos(1/4)^6 + sin(1/4)^6) - 10)/6. (End)

A073449 Decimal expansion of cot(1).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 01 2002

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

			0.64209261593433070300641998659...

Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, pp. 413-414.
Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.
Index entries for transcendental numbers

Cf. A049471 (tan(1)=1/A073449), A049469 (sin(1)), A049470 (cos(1)), A073447 (csc(1)), A073448 (sec(1)).

RealDigits[Cot[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)

PARI
```
cotan(1)
```

Equals Sum_{k>=0} (-1)^k * B(2*k) * 2^(2*k) / (2*k)!, where B(k) is the k-th Bernoulli number. - Amiram Eldar, May 15 2021

A073447 Decimal expansion of csc(1).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 01 2002

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

			1.18839510577812121626159945237...

Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395. Solution published in Vol. 37 , No. 5, November 2006, pp. 394-395.
Index entries for transcendental numbers.

Cf. A049469 (sin(1)=1/A073447), A049470 (cos(1)), A049471 (tan(1)), A073448 (sec(1)), A073449 (cot(1)).

Cf. A027641, A027642.

RealDigits[Csc[1], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

PARI
```
1/sin(1)
```

Equals Sum_{n=-oo..oo} ((-1)^n/(1 + n*Pi)). - Jean-François Alcover, Mar 21 2013.

Equals Sum_{k>=0} (-1)^k * (2 - 4^k) * bernoulli(2*k)/(2*k)! = Sum_{k>=0} (-1)^k * (2 - 4^k) * A027641(2*k)/(A027642(2*k)*(2*k)!). - Amiram Eldar, Aug 03 2020

A222480 Decimal expansion of cos(1)/(1+cos(1)).

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Feb 21 2013

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

			0.35077679479523758155811675...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A222481 (continued fraction), A222482 (Engel expansion), A049470 (cos(1)), A073448 (1/cos(1)).

s:= convert(evalf(1/(1+1/cos(1)), 140), string):
seq(parse(s[n+2]), n=0..122);

1/(1+1/cos(1)) \\ Charles R Greathouse IV, May 13 2019

cos(1)/(1+cos(1)) = 1/(1+1/cos(1)) = 1/(1+sec(1)).

A222481 Continued fraction for cos(1)/(1+cos(1)).

Original entry on oeis.org

0, 2, 1, 5, 1, 2, 2, 1, 2, 1, 1, 40, 4, 3, 1, 3, 4, 46, 3, 5, 2, 12, 3, 1, 4, 11, 1, 3, 3, 106, 2, 1, 1, 2, 28, 1, 7, 32, 3, 42, 1, 9, 1, 4, 2, 2, 1, 1, 15, 1, 5, 1, 8, 1, 1, 1, 3, 1, 7, 18, 13, 1, 1, 1, 1, 1, 5, 9, 7, 3, 3, 1, 1, 18, 6, 5, 1, 11, 1, 1, 10, 2

Offset: 0

Alois P. Heinz, Feb 21 2013

Essentially the same as A068115: a(n) = A068115(n) for n != 2.

			0.35077679479523758155811675...

Cf. A222480 (decimal expansion), A222482 (Engel expansion), A049470 (cos(1)), A073448 (1/cos(1)), A068115.

numtheory[cfrac](evalf(1/(1+1/cos(1)), 250), 130, 'quotients')[];

cos(1)/(1+cos(1)) = 1/(1+1/cos(1)) = 1/(1+sec(1)).

Offset changed by Andrew Howroyd, Jul 03 2024

A222482 Engel expansion of cos(1)/(1+cos(1)).

Original entry on oeis.org

3, 20, 22, 40, 68, 248, 7163, 28663, 50059, 64574, 638169, 761733, 2537764, 2925739, 3363073, 4977902, 5646039, 57212854, 159650555, 219684539, 453524713, 459239955, 2002180165, 3234082460, 14965375439, 50298730245, 89316768769, 464076054936, 520232391320

Offset: 1

Alois P. Heinz, Feb 21 2013

nonn

Cf. A006784 for definition of Engel expansion.

			0.35077679479523758155811675...

Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A222480 (decimal expansion), A222481 (continued fraction), A049470 (cos(1)), A073448 (1/cos(1)).

Digits:= 1000:
b:= evalf(1/(1+1/cos(1))):  engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
engel(b, 39);

cos(1)/(1+cos(1)) = 1/(1+1/cos(1)) = 1/(1+sec(1)).

A346839 Decimal expansion of Sum_{n>=0} A346838(n) / n!.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Aug 13 2021

			0.29340799302602338740477843394029002038314588271248926...

Cf. A049471, A073448, A346838.

A := n -> I*(polylog(-n, -I) - exp(I*Pi*n)*polylog(-n, I)) / exp(I*Pi*n/2):
Digits := 120; sum(A(n)/n!, n = 0..infinity): evalf(%)*10^91:
ListTools:-Reverse(convert(floor(%), base, 10));

RealDigits[Sec[1] - Tan[1], 10 , 100][[1]] (* Amiram Eldar, Aug 17 2021 *)

Equals sec(1) - tan(1) = A073448 - A049471. - Amiram Eldar, Aug 17 2021

A117041 Decimal expansion of (cos 1)^(-2)=(sec(1))^2.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Apr 16 2006

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

			3.42551882081475976094167893354113664805374743205738476586851476

Index entries for transcendental numbers

Cf. A049470, A073448.

RealDigits[1/Cos[1]^2,10,120][[1]] (* Harvey P. Dale, Jul 27 2021 *)

cos(1)^-2 \\ Charles R Greathouse IV, May 13 2019

A117042 Decimal expansion of (cos 1)^(-3)=(sec(1))^3.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Apr 16 2006

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

			6.340004074775787630861652711841658616429223645815934292875740623000

Index entries for transcendental numbers

Cf. A049470, A073448.

RealDigits[1/Cos[1]^3,10,120][[1]] (* Harvey P. Dale, Apr 29 2018 *)

cos(1)^-3 \\ Charles R Greathouse IV, May 13 2019

A117043 Decimal expansion of (cos 1)^(-4)=(sec(1))^4.

Original entry on oeis.org

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

Offset: 2

Mohammad K. Azarian, Apr 16 2006

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

			11.7341791917561421904326776865351274688644805049446095726008918896

Index entries for transcendental numbers

Cf. A049470, A073448.

RealDigits[Sec[1]^4,10,120][[1]] (* Harvey P. Dale, Jul 17 2021 *)

cos(1)^-4 \\ Charles R Greathouse IV, May 13 2019

cp's OEIS Frontend

A049470 Decimal expansion of cos(1).

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A073449 Decimal expansion of cot(1).

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A073447 Decimal expansion of csc(1).

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A222480 Decimal expansion of cos(1)/(1+cos(1)).

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A222481 Continued fraction for cos(1)/(1+cos(1)).

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A222482 Engel expansion of cos(1)/(1+cos(1)).

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A346839 Decimal expansion of Sum_{n>=0} A346838(n) / n!.

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