A019774 Decimal expansion of sqrt(e).

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

Offset

1,2

Comments

Also where x^(x^(-2)) is a maximum. - Robert G. Wilson v, Oct 22 2014

e^(1/2) maximizes the value of x^(c/(x^2)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

Michael Penn, On means of binomial coefficients., YouTube video, 2020.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 522.

Jonathan Sondow and Huang Yi, New Wallis- and Catalan-Type Infinite Products for Pi, e and sqrt(2+sqrt(2)), The American Mathematical Monthly, Vol. 117, No. 10 (2010), pp. 912-917.

Index entries for transcendental numbers.

Formula

sqrt(e) = Sum_{n>=0} 1/(2^n*n!) = Sum_{n>=0} 1/(2n)!!. - Daniel Forgues, Apr 17 2011

sqrt(e) = 1 + Sum_{n>0} Product_{i=1..n} 1/(2n). - Ralf Stephan, Sep 11 2013

Continued fraction representation: sqrt(e) = 1 + 1/(1 + 2/(3 + 4/(5 + ... ))). See A000354 for details. - Peter Bala, Jan 30 2015

sqrt(e) = (1/2)*( 1 + (3 + (5 + (7 + ...)/6)/4)/2 ) = 1 + (1 + (1 + (1 + ...)/6)/4)/2. - Rok Cestnik, Jan 19 2017

sqrt(e) = 2*Sum_{n >= 0} 1/((1 - 4*n^2)*(2^n)*n!). - Peter Bala, Jan 16 2022

sqrt(e) = (16/31)*(1 + Sum_{n>=1} (1/2)^n*((1/2)*n^3 + (1/2)*n + 1)/n!). - Alexander R. Povolotsky, Jul 01 2022

sqrt(e) = Sum_{n >= 0} (n + 1/2)/(2^n*n!). - Peter Bala, Jun 29 2024

Equals i^(-i/Pi), where i denotes the imaginary unit. - Stefano Spezia, Mar 01 2025

Equals 2 * Product_{k>=1} (2^(2^(k-1))*(2^k-1)!!^3/((2^(k-1)-1)!!^2*(2^(k+1)-1)!!))^(1/2^k) = 2 * (2/3)^(1/2) * (6*6/(5*7))^(1/4) * (10*10*14*14/(9*11*13*15))^(1/8) * ... (Sondow and Yi, 2010, p. 914, eq. (16)). - Amiram Eldar, Jul 02 2026

Example

1.6487212707001281468486507878141635716537761007101480115750...

Maple

evalf(sqrt(exp(1)), 120); # Muniru A Asiru, Aug 16 2018

Mathematica

RealDigits[N[Sqrt[E], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

Prog

(PARI) default(realprecision, 20080); x=sqrt(exp(1)); for (n=1, 200, d=floor(x); x=(x-d)*10; write("b019774.txt", n, " ", d)); \\ Harry J. Smith, May 01 2009
(Magma)
m:= 510; R:=RealField(m); SetDefaultRealField(R);
Prune(Reverse(IntegerToSequence(Floor(( Exp(1/2) )*10^(Floor(m/2)) )))); // G. C. Greubel, Nov 06 2025
(SageMath)
numerical_approx( exp(1/2) , digits=265 ) # G. C. Greubel, Nov 06 2025

Crossrefs

Cf. A000354, A001113, A058281 for continued fraction for sqrt(e).

Cf. A019775, A019776, A019777, A019778, A019779, A019780, A019781, A019782, A019784, A019785, A019786, A019787, A019788, A019789, A019790, A019791 A019792, A019794, A019795, A019796, A019797, A019798.

Sequence in context: A388500 A176786 A077669 * A354630 A254250 A195434

Adjacent sequences: A019771 A019772 A019773 * A019775 A019776 A019777

Keyword

nonn,cons,changed

Author

N. J. A. Sloane

Status

approved