OFFSET
0,5
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]
Keith Matthews, Finding the continued fraction of e^(l/m).
Thomas J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
Gang Xiao, Contfrac.
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
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + 2*log(sqrt(2)+1)) / (4*sqrt(2)). - Amiram Eldar, May 03 2025
EXAMPLE
sqrt(e) = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(5 + ...)))). - Harry J. Smith, May 01 2009
MATHEMATICA
ContinuedFraction[ Sqrt[E], 100]
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 1, 1, 1, 5, 1}, 100] (* Harvey P. Dale, Aug 05 2025 *)
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
KEYWORD
cofr,nonn,easy,nice
AUTHOR
Robert G. Wilson v, Dec 07 2000
EXTENSIONS
More terms from Jason Earls, Jul 10 2001
STATUS
approved
