A049469 -id:A049469 - 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)

A073742 Decimal expansion of sinh(1).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 07 2002

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

Decimal expansion of u > 0 such that 1 = arclength on the hyperbola y^2 - x^2 = 1 from (0,0) to (u,y(u)). - Clark Kimberling, Jul 04 2020

			1.17520119364380145688238185059...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:7 at page 20.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Hyperbolic Sine.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Eric Weisstein's World of Mathematics, Factorial Sums.
Index entries for transcendental numbers.

Cf. A068139 (continued fraction), A073743, A073744, A073745, A073746, A073747, A049469, A049470, A174548.

First@ RealDigits@ N[Sinh@ 1, 120] (* Michael De Vlieger, Sep 04 2016 *)

PARI
```
sinh(1)
```

Equals (e - e^(-1))/2.
Equals sin(i)/i. - N. J. A. Sloane, Feb 12 2010
Equals Sum_{n>=0} 1/A009445(n). See Gradsteyn-Ryzhik (0.245.6.) - R. J. Mathar, Oct 27 2012
Continued fraction representation: sinh(1) = 1 + 1/(6 - 6/(21 - 20/(43 - 42/(73 - ... - (2*n - 1)*(2*n - 2)/((2*n*(2*n + 1) + 1) - ... ))))). See A051397 for proof. Cf. A049469. - Peter Bala, Sep 02 2016
Equals Product_{k>=1} 1 + 1/(k * Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073745 = A174548/2. - Hugo Pfoertner, Dec 27 2024

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

A073448 Decimal expansion of sec(1).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 01 2002

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

			1.85081571768092561791175324139...

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. A049470 (cos(1)=1/A073448), A049469 (sin(1)), A049471 (tan(1)), A073447 (csc(1)), A073449 (cot(1)), A122045.

RealDigits[Sec[1],10,120][[1]] (* Harvey P. Dale, Mar 13 2013 *)

PARI
```
1/cos(1)
```

Equals Sum_{k>=0} (-1)^k * E(2*k) / (2*k)!, where E(k) is the k-th Euler number (A122045). - Amiram Eldar, May 15 2021

A143623 Decimal expansion of the constant cos(1) + sin(1).

Original entry on oeis.org

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

Offset: 1

Peter Bala, Aug 30 2008

cos(1) + sin(1) = Sum_{n >= 0} (-1)^floor(n/2)/n! = 1 + 1/1! - 1/2! - 1/3! + 1/4! + 1/5! - 1/6! - 1/7! + + - - ... .
Define E_2(k) = Sum_{n >= 0} (-1)^floor(n/2)*n^k/n! for k = 0, 1, 2, ... . Then E_2(0) = cos(1) + sin(1) and E_2(1) = cos(1) - sin(1).
Furthermore, E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = E_2(1) - E_2(0), E_2(3) = -3*E_2(0) and E_2(4) = -5*E_2(1) - 6*E_2(0). The precise result is E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1).
The decimal expansion of the constant cos(1) - sin(1) = E_2(1) is recorded in A143624. Compare with A143625.

			1.38177329067603622405 ... .

Cf. A049469, A049470, A057077, A121867, A121868, A143624, A143625.

RealDigits[Cos[1]+Sin[1],10,120][[1]] (* Harvey P. Dale, Mar 01 2019 *)

sqrt(2)*sin(1+Pi/4) \\ Charles R Greathouse IV, Feb 03 2025

Equals sin(1+Pi/4)*sqrt(2). - Franklin T. Adams-Watters, Jun 27 2014

Offset corrected by R. J. Mathar, Feb 05 2009

A143624 Decimal expansion of the negated constant cos(1) - sin(1) = -0.3011686789...

Original entry on oeis.org

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

Offset: 0

Peter Bala, Aug 30 2008

cos(1) - sin(1) = Sum_{n>=0} (-1)^floor(n/2)*n/n! = 1/1! - 2/2! - 3/3! + 4/4! + 5/5! - 6/6! - 7/7! + + - - ... . Define E_2(k) = Sum_{n>=0} (-1)^floor(n/2)*n^k/n! for k = 0,1,2,... . Then E_2(1) = cos(1) - sin(1) and E_2(0) = cos(1) + sin(1). Furthermore, E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = E_2(1) - E_2(0), E_2(3) = -3*E_2(0) and E_2(4) = -5*E_2(1) - 6*E_2(0). The precise result is E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1). The decimal expansion of the constant cos(1) + sin(1) is recorded in A143623. Compare with A143625.

			 -0.30116867893975678925156571418732239589025264018...

Eric Weisstein's World of Mathematics, Spherical Bessel Function of the First Kind

Cf. A049469, A049470, A057077, A121867, A121868, A143623, A143625, A174549.

RealDigits[Cos[1] - Sin[1], 10, 100][[1]] (* Amiram Eldar, Aug 07 2020 *)

cos(1)-sin(1) \\ Charles R Greathouse IV, Feb 04 2025

Equals sin(1-Pi/4)*sqrt(2). - Franklin T. Adams-Watters, Jun 27 2014
Equals j_1(1), where j_1(z) is the spherical Bessel function of the first kind. - Stanislav Sykora, Jan 11 2017
From Amiram Eldar, Aug 07 2020: (Start)
Equals -Integral_{x=0..1} x*sin(x) dx.
Equals Sum_{k>=1} (-1)^k/((2*k-1)! * (2*k+1)) = Sum_{k>=1} (-1)^k/A174549(k). (End)

Added sign in definition. Offset corrected by R. J. Mathar, Feb 05 2009

A354332 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k+1)!.

Original entry on oeis.org

1, 5, 101, 4241, 305353, 33588829, 209594293, 1100370038249, 23023126954133, 102360822438075317, 42991545423991633141, 4350744396907953273869, 13052233190723859821607001, 9162667699888149594768114701, 7440086172309177470951709137213, 364172638960396581472899447242531

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 5/6, 101/120, 4241/5040, 305353/362880, 33588829/39916800, 209594293/249080832, ...

Cf. A009445, A049469, A053557, A061354, A103816, A120265, A143382, A354211, A354298, A354333 (denominators), A354334.

Table[Sum[(-1)^k/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Sin[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Numerator

a(n) = numerator(sum(k=0, n, (-1)^k/(2*k+1)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354332(n): return sum(Fraction(-1 if k % 2 else 1,factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

Numerators of coefficients in expansion of sin(sqrt(x)) / (sqrt(x) * (1 - x)).

A354333 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (2*k+1)!.

Original entry on oeis.org

1, 6, 120, 5040, 362880, 39916800, 249080832, 1307674368000, 27360571392000, 121645100408832000, 51090942171709440000, 5170403347776995328000, 15511210043330985984000000, 10888869450418352160768000000, 8841761993739701954543616000000, 432780981798838043038187520000000

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 5/6, 101/120, 4241/5040, 305353/362880, 33588829/39916800, 209594293/249080832, ...

Cf. A009445, A049469, A053556, A061355, A143383, A354299, A354331, A354332 (numerators), A354335.

Table[Sum[(-1)^k/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Sin[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Denominator

a(n) = denominator(sum(k=0, n, (-1)^k/(2*k+1)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354333(n): return sum(Fraction(-1 if k % 2 else 1,factorial(2*k+1)) for k in range(n+1)).denominator # Chai Wah Wu, May 24 2022

Denominators of coefficients in expansion of sin(sqrt(x)) / (sqrt(x) * (1 - x)).

A074790 a(n) = (2n+1)!Sum_{k=0..n} (-1)^k/(2*k+1)!.

Original entry on oeis.org

1, 5, 101, 4241, 305353, 33588829, 5239857325, 1100370038249, 299300650403729, 102360822438075317, 42991545423991633141, 21753721984539766369345, 13052233190723859821607001, 9162667699888149594768114701

Offset: 0

Benoit Cloitre, Sep 07 2002

Seiichi Manyama, Table of n, a(n) for n = 0..224

Cf. A000166, A009551, A049469, A051397.

Table[(2n+1)!Sum[(-1)^k/(2k+1)!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 14 2019 *)

a(n) = (2*n+1)!*sum(k=0,n,(-1)^k/(2*k+1)!); \\ Michel Marcus, Sep 09 2016

[factorial(2*n+1)*sum((-1)^j/factorial(2*j+1) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 09 2021

a(n) = round(sin(1)*(2*n+1)!).
a(n) = A009551(2*n+1).
From Peter Bala, Jan 30 2015: (Start)
G.f.: sin(x)/(1 - x^2) = x + 5*x^3/3! + 101*x^5/5! + 4241*x^7/7! + ....
a(n) = 2*n*(2*n + 1)*a(n-1) + (-1)^n with a(0) = 1.
a(n) = (4*n^2 + 2*n - 1)*a(n-1) + (2*n-1)*(2*n-2)*a(n-2) with a(0) = 1, a(1) = 5.
The sequence b(n) := (2*n + 1)! also satisfies the second recurrence but with b(0) = 1, b(1) = 6. This leads to the continued fraction representation a(n) = (2*n + 1)!*(1 - 1/(6 + 6/(19 + 20/(41 + ... + (2*n - 1)*(2*n - 2)/(4*n^2 + 2*n - 1) )))) for n >= 2. Taking the limit gives the continued fraction representation sin(1) = 1 - 1/(6 + 6/(19 + 20/(41 + ... + (2*n - 1)*(2*n - 2)/((4*n^2 + 2*n - 1) + ... )))). (End)

cp's OEIS Frontend

A049470 Decimal expansion of cos(1).

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A073742 Decimal expansion of sinh(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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A073448 Decimal expansion of sec(1).

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A143623 Decimal expansion of the constant cos(1) + sin(1).

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A143624 Decimal expansion of the negated constant cos(1) - sin(1) = -0.3011686789...

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A074790 a(n) = (2n+1)!Sum_{k=0..n} (-1)^k/(2*k+1)!.