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

Author

Peter Bala, Aug 30 2008

Keywords

• cons
• easy
• nonn

Comments

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.

Examples

			1.38177329067603622405 ... .
		

Crossrefs

Cf. A049469, A049470, A057077, A121867, A121868, A143624, A143625.

Programs

Mathematica

RealDigits[Cos[1]+Sin[1],10,120][[1]] (* Harvey P. Dale, Mar 01 2019 *)

PARI

sqrt(2)*sin(1+Pi/4) \\ Charles R Greathouse IV, Feb 03 2025

Formula

Equals sin(1+Pi/4)*sqrt(2). - Franklin T. Adams-Watters, Jun 27 2014

Extensions

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