A143625 -id:A143625 - cp's OEIS frontend

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

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

A143626 Decimal expansion of the constant E_3(1) := Sum_{k >= 0} (-1)^floor(k/3)*k/k! = 1/1! + 2/2! - 3/3! - 4/4! - 5/5! + + + - - - ... .

Original entry on oeis.org

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

Offset: 1

Peter Bala, Aug 30 2008

Define E_3(n) = Sum_{k >= 0} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E_3(n+3) = 3*E_3(n+2) - 2*E_3(n+1) - Sum_{i = 0..n} 3^i*binomial(n,i) * E_3(n-i) for n >= 0. Thus E_3(n) is an integral linear combination of E_3(0), E_3(1) and E_3(2). See the examples below.
The decimal expansions of E_3(0) and E_3(2) are given in A143635 and A143627. Compare with A143623 and A143624.
E_3(n) as linear combination of E_3(i), i = 0..2.
=======================================
..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)
=======================================
..E_3(3)..|.....-1.......-2........3...
..E_3(4)..|.....-6.......-7........7...
..E_3(5)..|....-25......-23.......14...
..E_3(6)..|....-89......-80.......16...
..E_3(7)..|...-280.....-271......-77...
..E_3(8)..|...-700.....-750.....-922...
..E_3(9)..|...-380.....-647....-6660...
..E_3(10).|..13452....13039...-41264...
...
The columns are A143628, A143629 and A143630.

			1.3015594959829796028430427

Cf. A143623, A143624, A143625, A143627, A143628, A143629, A143630.

RealDigits[ N[ (4*E^(3/2)*Cos[Sqrt[3]/2] - 1)/(3*E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

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

A143627 Decimal expansion of the constant E_3(2) := sum {k = 0.. inf} (-1)^floor(k/3)*k^2/k! = 1/1! + 2^2/2! - 3^2/3! - 4^2/4! - 5^2/5! + + + - - - ... = 0.68605 60507 ... .

Original entry on oeis.org

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

Offset: 0

Peter Bala, Aug 30 2008

Define E_3(n) = sum {k = 0..inf} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E_3(n+3) = 3*E_3(n+2) - 2*E_3(n+1) - sum {i = 0..n} 3^i*binomial(n,i) * E_3(n-i) for n >= 0. Thus E_3(n) is an integral linear combination of E_3(0), E_3(1) and E_3(2). See the examples below. The decimal expansions of E_3(0) and E_3(1) are given in A143625 and A143626. Compare with A143623 and A143624.

			E_3(n) as linear combination of E_3(i),
i = 0..2.
=======================================
..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)
=======================================
..E_3(3)..|.....-1.......-2........3...
..E_3(4)..|.....-6.......-7........7...
..E_3(5)..|....-25......-23.......14...
..E_3(6)..|....-89......-80.......16...
..E_3(7)..|...-280.....-271......-77...
..E_3(8)..|...-700.....-750.....-922...
..E_3(9)..|...-380.....-647....-6660...
..E_3(10).|..13452....13039...-41264...
...
The columns are A143628, A143629 and A143630.

A143623, A143624, A143625, A143626, A143628, A143629, A143630.

RealDigits[N[(8/3)*Sqrt[E]*Cos[Sqrt[3]/2] + (1/40)*(HypergeometricPFQ[{}, {7/3, 8/3}, -(1/27)] - 5*HypergeometricPFQ[{}, {5/3, 7/3}, -(1/27)]) - 2*Sqrt[E/3]*Sin[Sqrt[3]/2] - 5/(3*E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A143626 Decimal expansion of the constant E_3(1) := Sum_{k >= 0} (-1)^floor(k/3)*k/k! = 1/1! + 2/2! - 3/3! - 4/4! - 5/5! + + + - - - ... .

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A143627 Decimal expansion of the constant E_3(2) := sum {k = 0.. inf} (-1)^floor(k/3)*k^2/k! = 1/1! + 2^2/2! - 3^2/3! - 4^2/4! - 5^2/5! + + + - - - ... = 0.68605 60507 ... .

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions