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%I A001113 M1727 N0684
%S A001113 2,7,1,8,2,8,1,8,2,8,4,5,9,0,4,5,2,3,5,3,6,0,2,8,7,4,7,1,3,5,2,6,6,2,4,
%T A001113 9,7,7,5,7,2,4,7,0,9,3,6,9,9,9,5,9,5,7,4,9,6,6,9,6,7,6,2,7,7,2,4,0,7,6,
%U A001113 6,3,0,3,5,3,5,4,7,5,9,4,5,7,1,3,8,2,1,7,8,5,2,5,1,6,6,4,2,7,4,2,7,4,6
%N A001113 Decimal expansion of e.
%C A001113 e is sometimes called Euler's constant, also Napier's constant.
%C A001113 Also, decimal expansion of sinh(1)+cosh(1) - Mohammad K. Azarian (azarian(AT)evansville.edu), 
               Aug 15 2006
%C A001113 If m and n are any integers with n > 1, then |e - m/n| > 1/(S(n)+1)!, 
               where S(n) = A002034(n) is the smallest number such that n divides 
               S(n)!. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Sep 04 
               2006
%D A001113 Mohammad K. Azarian, An Expansion of e, Problem # B-765, Fibonacci Quarterly, 
               Vol. 32, No. 2, May 1994, p. 181. Solution appeared in Vol. 33, No. 
               4, Aug.1995, p. 377. [From Mohammad K. Azarian (azarian(AT)evansville.edu), 
               Feb 08 2009]
%D A001113 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.
%D A001113 E. Maor, e: The Story of a Number, Princton Univ. Press, 1994.
%D A001113 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 
               52.
%D A001113 G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 
               decimal places. Math. Tables and Other Aids to Computation 4, (1950). 
               11-15.
%D A001113 D. Shanks and J. W. Wrench, Jr., Calculation of e to 100,000 decimals, 
               Math. Comp., 23 (1969), 679-680.
%D A001113 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001113 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001113 J. Sondow, A geometric proof that e is irrational and a new measure of 
               its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
%H A001113 N. J. A. Sloane, <a href="b001113.txt">Table of 50000 digits of e labeled 
               from 1 to 50000</a> [based on the ICON Project link below]
%H A001113 Dave's Math Tables, <a href="http://www.math2.org/math/constants/e.htm">
               e</a>
%H A001113 X. Gourdon, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/
               expof1.txt">e to 1.250 billion digits</a>
%H A001113 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/
               Constants/E/e.html">The constant e and its computation</a>
%H A001113 ICON Project, <a href="http://www.cs.arizona.edu/icon/oddsends/e.htm">
               e to 50000 places</a>
%H A001113 R. Nemiroff and J. Bonnell, <a href="http://antwrp.gsfc.nasa.gov/htmltest/
               gifcity/e.5mil">The first 5 million digits of the number e</a>
%H A001113 J. J. O'Connor & E. F. Robertson, <a href="http://www-groups.dcs.st-and.ac.uk/
               ~history/HistTopics/e.html">The number e</a>
%H A001113 S. Plouffe, <a href="http://www.cecm.sfu.ca/projects/ISC/records.html">
               A million digits</a>
%H A001113 E. Sandifer, How Euler Did It, <a href="http://www.maa.org/editorial/
               euler/How%20Euler%20Did%20It%2028%20e%20is%20irrational.pdf">Who 
               proved e is irrational?</a>
%H A001113 Jean-Louis Sigrist, <a href="http://jlsigrist.com/e.html">Le premier 
               million de decimales de e</a>. [From Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Sep 28 2009]
%H A001113 G. Villemin's Almanach of Numbers, <a href="http://translate.google.com/
               translate?hl=en&sl=fr&u=http://membres.lycos.fr/villemingerard/Analyse/
               ExpoVal.htm">Constant"e"</a>
%H A001113 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               e.html">Link to a section of The World of Mathematics.</a>
%H A001113 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               UniformSumDistribution.html">Uniform Sum Distribution</a>
%H A001113 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FactorialSums.html">Factorial Sums</a>
%H A001113 Wikipedia, <a href="http://www.wikipedia.org/wiki/E_%28mathematical_constant%29">
               E(mathematical constant</a>
%F A001113 e = Sum_{k >= 0} 1/k! = lim_{x -> 0} (1+x)^(1/x).
%F A001113 e is the unique positive root of the equation Integral_{u = 1..x} du/
               u = 1.
%F A001113 exp(1)= (16/31*(sum((1/2)^n*(1/2*n^3+1/2*n+1)/n!,n=1..infinity) +1))^2. 
               Robert Israel confirmed that above formula is correct, saying: "In 
               fact, sum(n^j*t^n/n!, n=0..infinity) = P_j(t)*exp(t) where P_0(t) 
               = 1 and for j >= 1, P_j(t) = t (P_(j-1)'(t) + P_(j-1)(t)). Your sum 
               is 1/2*P_3(1/2) + 1/2*P_1(1/2) + P_0(1/2) ." [From Alexander R. Povolotsky 
               (pevnev(AT)juno.com), Jan 04 2009]
%e A001113 2.71828182845904523536028747135266249775724709369995957496696762772407663\
%e A001113 0353547594571382178525166427427466391932003059921817413596629043572900334\
%e A001113 295260595630738132328627943490763233829880753195251019...
%p A001113 Digits := 200: it := evalf((exp(1))/10, 200): for i from 1 to 200 do 
               printf(`%d,`,floor(10*it)): it := 10*it-floor(10*it): od:
%t A001113 a := N[E, 500]; For[n = 1, n < 250, n++, Print[Floor[10^(n - 1)*a] - 
               Floor[10^(n - 2)*a]*10]] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Feb 17 2006
%o A001113 (PARI) { default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); 
               x=(x-d)*10; write("b001113.txt", n, " ", d)); } [From Harry J. Smith 
               (hjsmithh(AT)sbcglobal.net), Apr 15 2009]
%Y A001113 Cf. A002034, A122214, A122215, A122216, A122217, A122416, A122417.
%Y A001113 Sequence in context: A021372 A111714 A060302 this_sequence A094121 A105178 
               A112257
%Y A001113 Adjacent sequences: A001110 A001111 A001112 this_sequence A001114 A001115 
               A001116
%K A001113 nonn,cons,nice,core
%O A001113 1,1
%A A001113 N. J. A. Sloane (njas(AT)research.att.com).
%E A001113 Maple code from James A. Sellers (sellersj(AT)math.psu.edu), Feb 13 2001

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