A267318 Continued fraction expansion of e^(1/5).

1, 4, 1, 1, 14, 1, 1, 24, 1, 1, 34, 1, 1, 44, 1, 1, 54, 1, 1, 64, 1, 1, 74, 1, 1, 84, 1, 1, 94, 1, 1, 104, 1, 1, 114, 1, 1, 124, 1, 1, 134, 1, 1, 144, 1, 1, 154, 1, 1, 164, 1, 1, 174, 1, 1, 184, 1, 1, 194, 1, 1, 204, 1, 1, 214, 1, 1, 224, 1, 1, 234, 1, 1, 244, 1, 1, 254, 1, 1, 264, 1, 1

Offset

0,2

Comments

e^(1/5) is a transcendental number.

In general, the ordinary generating function for the continued fraction expansion of e^(1/k), with k = 1, 2, 3..., is (1 + (k - 1)*x + x^2 - (k + 1)*x^3 + 7*x^4 - x^5)/(1 - x^3)^2.

Links

Table of n, a(n) for n=0..81.

Eric Weisstein's World of Mathematics, e Continued Fraction

Index entries for continued fractions for constants

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

Formula

G.f.: (1 + 4*x + x^2 - x^3 + 6*x^4 - x^5)/(1 - x^3)^2.

a(n) = 1 + (3 + 10*floor(n/3))*(1 - (n-1)^2 mod 3). [Bruno Berselli, Feb 04 2016]

Example

e^(1/5) = 1 + 1/(4 + 1/(1 + 1/(1 + 1/(14 + 1/(1 + 1/...))))).

Mathematica

ContinuedFraction[Exp[1/5], 82]
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 4, 1, 1, 14, 1}, 82]
CoefficientList[Series[(1 + 4 x + x^2 - x^3 + 6 x^4 - x^5) / (x^3 - 1)^2, {x, 0, 70}], x] (* Vincenzo Librandi, Jan 13 2016 *)
Table[1 + (3 + 10 Floor[n/3]) (1 - Mod[(n - 1)^2, 3]), {n, 0, 90}] (* Bruno Berselli, Feb 04 2016 *)

Prog

(Magma) [1+(3+10*Floor(n/3))*(1-(n-1)^2 mod 3): n in [0..90]]; // Bruno Berselli, Feb 04 2016

Crossrefs

Cf. A092514.

Cf. continued fraction expansion of e^(1/k): A003417 (k=1), A058281 (k=2), A078689 (k=3), A078688 (k=4), this sequence (k=5).

Sequence in context: A022167 A339849 A064281 * A050154 A179454 A058711

Adjacent sequences: A267315 A267316 A267317 * A267319 A267320 A267321

Keyword

nonn,cofr,easy

Author

Ilya Gutkovskiy, Jan 13 2016

Extensions

Edited by Bruno Berselli, Feb 04 2016

Status

approved