A058281 Continued fraction for square root of e.

1, 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, 21, 1, 1, 25, 1, 1, 29, 1, 1, 33, 1, 1, 37, 1, 1, 41, 1, 1, 45, 1, 1, 49, 1, 1, 53, 1, 1, 57, 1, 1, 61, 1, 1, 65, 1, 1, 69, 1, 1, 73, 1, 1, 77, 1, 1, 81, 1, 1, 85, 1, 1, 89, 1, 1, 93, 1, 1, 97, 1, 1, 101, 1, 1, 105, 1, 1, 109, 1, 1, 113, 1, 1

Offset

0,5

Comments

sqrt(e) = 1.64872127070012814684865078781416357165377610071014801157507... - Harry J. Smith, May 01 2009

Links

Harry J. Smith, Table of n, a(n) for n = 0..20000

D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]

K. Matthews, Finding the continued fraction of e^(l/m)

T. J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.

G. Xiao, Contfrac

Index entries for continued fractions for constants

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Formula

a(3k+1) = 4k+1, a(i) = 1 otherwise.

G.f.: -(x^2-x+1)*(x^3-2*x^2-2*x-1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Jun 24 2013

E.g.f.: exp(-x/2)*(exp(3*x/2)*(5 + 4*x) + (4 + 8*x)*cos(sqrt(3)*x/2) - 4*sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, May 05 2022

Example

sqrt(e) = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(5 + ...)))). - Harry J. Smith, May 01 2009

Mathematica

ContinuedFraction[ Sqrt[E], 100]

Prog

(PARI) contfrac(sqrt(exp(1)))
(PARI) { allocatemem(932245000); default(realprecision, 60000); x=contfrac(sqrt(exp(1))); for (n=1, 20001, write("b058281.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 01 2009

Crossrefs

Cf. A004766, A016813.

Cf. A019774 for decimal expansion of sqrt(e).

Sequence in context: A054110 A132048 A141398 * A046583 A046579 A153108

Adjacent sequences: A058278 A058279 A058280 * A058282 A058283 A058284

Keyword

cofr,nonn,easy,nice

Author

Robert G. Wilson v, Dec 07 2000

Extensions

More terms from Jason Earls, Jul 10 2001

Status

approved