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%I #23 Aug 07 2020 10:04:39
%S 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,
%T 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,
%U 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
%N Decimal expansion of the negated constant cos(1) - sin(1) = -0.3011686789...
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SphericalBesselFunctionoftheFirstKind.html">Spherical Bessel Function of the First Kind</a>
%F Equals sin(1-Pi/4)*sqrt(2). - _Franklin T. Adams-Watters_, Jun 27 2014
%F Equals j_1(1), where j_1(z) is the spherical Bessel function of the first kind. - _Stanislav Sykora_, Jan 11 2017
%F From _Amiram Eldar_, Aug 07 2020: (Start)
%F Equals -Integral_{x=0..1} x*sin(x) dx.
%F Equals Sum_{k>=1} (-1)^k/((2*k-1)! * (2*k+1)) = Sum_{k>=1} (-1)^k/A174549(k). (End)
%e cos(1) - sin(1) = -0.30116867893975678925156571418732239589025264018...
%t RealDigits[Cos[1] - Sin[1], 10, 100][[1]] (* _Amiram Eldar_, Aug 07 2020 *)
%Y Cf. A049469, A049470, A057077, A121867, A121868, A143623, A143625, A174549.
%K cons,easy,nonn
%O 0,1
%A _Peter Bala_, Aug 30 2008
%E Added sign in definition. Offset corrected by _R. J. Mathar_, Feb 05 2009
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