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

1,2

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.

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

Example

1.359140914229522617680143735676331248878623546849979787483483813862038... = A001113/2.

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 *)

Prog

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

Crossrefs

Cf. A001113, A006083 (continued fraction).

Sequence in context: A258086 A141251 A186190 * A101298 A296452 A225594

Adjacent sequences: A019736 A019737 A019738 * A019740 A019741 A019742

Keyword

nonn,cons

Author

N. J. A. Sloane

Status

approved