A019739 Decimal expansion of e/2.

1, 3, 5, 9, 1, 4, 0, 9, 1, 4, 2, 2, 9, 5, 2, 2, 6, 1, 7, 6, 8, 0, 1, 4, 3, 7, 3, 5, 6, 7, 6, 3, 3, 1, 2, 4, 8, 8, 7, 8, 6, 2, 3, 5, 4, 6, 8, 4, 9, 9, 7, 9, 7, 8, 7, 4, 8, 3, 4, 8, 3, 8, 1, 3, 8, 6, 2, 0, 3, 8, 3, 1, 5, 1, 7, 6, 7, 7, 3, 7, 9, 7, 2, 8, 5, 6, 9, 1, 0, 8, 9, 2, 6, 2, 5, 8, 3, 2, 1

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Examples

			1.359140914229522617680143735676331248878623546849979787483483813862038... = A001113/2.
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.3, p. 14.
• Jolley, Summation of Series, Dover (1961) eq. (161) on page 30.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• R. P. Millane., A product form of the Möbius transform, Whistler Center for Carbohydrate Research, Purdue University, West Lafayette, USA.
• Roger H. Moritz, Summing series, PRIMUS, 1 (2) (2007) 212-219, Comment 2.
• Index entries for transcendental numbers.

Crossrefs

Cf. A001113, A006083 (continued fraction).

Programs

Magma

SetDefaultRealField(RealField(100)); Exp(1)/2; // Vincenzo Librandi, Apr 05 2020

Mathematica

N[Product[((1/n)!)^MoebiusMu[n], {n, 1, 200000}]] (* John M. Campbell, Jun 14 2011 *)
RealDigits[E/2,10,120][[1]] (* Harvey P. Dale, Sep 18 2018 *)

PARI

default(realprecision, 20080); x=exp(1)/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019739.txt", n, " ", d)); \\ Harry J. Smith, May 10 2009

PARI

digits(10^default(realprecision)*exp(1)\20) \\ M. F. Hasler, Apr 01 2020

Formula

e/2 = lim_{n->oo} n*(e - (1+1/n)^n). - Benoit Cloitre, Sep 17 2002

e/2 = Product_{n>=1} ((1/n)!)^mu(n), where mu is the Mobius function is an unusual infinite product for this number: (see Millane ref.). - John M. Campbell, Jun 14 2011

10*(this constant) = 5*exp(1) = Sum_{j>=0} j^3/j! [Jolley]. - R. J. Mathar, Oct 03 2011

Equals Sum_{j>=0} (1+j)/(1+2*j)!. - Bruno Berselli, May 25 2015

Equals the coefficient of x in Sum_{m>1} log((1 - x/m!)(1 - 2x/m!)...(1 - (m-1)x/m!)). - M. F. Hasler, Apr 01 2020

Equals A001113/2 = Sum_{k>=1} k*(k-1)/(2 * k!). - Amiram Eldar, Aug 10 2020