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

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

Author

Peter Bala, Aug 30 2008

Keywords

• cons
• easy
• nonn

Comments

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.

Examples

			 -0.30116867893975678925156571418732239589025264018...
		

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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)

Extensions

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