|
%I #20 Oct 14 2023 11:32:39
%S 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,
%T 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,
%U 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
%N Decimal expansion of the constant cos(1) + sin(1).
%C 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! + + - - ... .
%C 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).
%C 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).
%C The decimal expansion of the constant cos(1) - sin(1) = E_2(1) is recorded in A143624. Compare with A143625.
%F Equals sin(1+Pi/4)*sqrt(2). - _Franklin T. Adams-Watters_, Jun 27 2014
%e 1.38177329067603622405 ... .
%t RealDigits[Cos[1]+Sin[1],10,120][[1]] (* _Harvey P. Dale_, Mar 01 2019 *)
%Y Cf. A049469, A049470, A057077, A121867, A121868, A143624, A143625.
%K cons,easy,nonn
%O 1,2
%A _Peter Bala_, Aug 30 2008
%E Offset corrected by _R. J. Mathar_, Feb 05 2009
|
