A078689 Continued fraction expansion of e^(1/3).

1, 2, 1, 1, 8, 1, 1, 14, 1, 1, 20, 1, 1, 26, 1, 1, 32, 1, 1, 38, 1, 1, 44, 1, 1, 50, 1, 1, 56, 1, 1, 62, 1, 1, 68, 1, 1, 74, 1, 1, 80, 1, 1, 86, 1, 1, 92, 1, 1, 98, 1, 1, 104, 1, 1, 110, 1, 1, 116, 1, 1, 122, 1, 1, 128, 1, 1, 134, 1, 1, 140, 1, 1, 146

Offset: 0

Author

Benoit Cloitre, Dec 17 2002

Keywords

• cofr
• nonn
• easy

Links

• Thomas J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A016933, A058281, A092041.

Programs

Mathematica

ContinuedFraction[Exp[1/3], 100] (* Amiram Eldar, May 20 2022 *)

Formula

a(3k+1) = 6k+2, otherwise a(i) = 1.

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

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/18 + log(2)/6. - Amiram Eldar, May 04 2025