OFFSET
0,1
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.
LINKS
Eric Weisstein's World of Mathematics, Spherical Bessel Function of the First Kind
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)
EXAMPLE
-0.30116867893975678925156571418732239589025264018...
MATHEMATICA
RealDigits[Cos[1] - Sin[1], 10, 100][[1]] (* Amiram Eldar, Aug 07 2020 *)
PROG
(PARI) cos(1)-sin(1) \\ Charles R Greathouse IV, Feb 04 2025
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Aug 30 2008
EXTENSIONS
Added sign in definition. Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved