A006083 Continued fraction for e/2. (Formerly M0125)

1, 2, 1, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 5, 3, 1, 5, 1, 3, 7, 3, 1, 7, 1, 3, 9, 3, 1, 9, 1, 3, 11, 3, 1, 11, 1, 3, 13, 3, 1, 13, 1, 3, 15, 3, 1, 15, 1, 3, 17, 3, 1, 17, 1, 3, 19, 3, 1, 19, 1, 3, 21, 3, 1, 21, 1, 3, 23, 3, 1, 23, 1, 3, 25, 3, 1, 25, 1, 3, 27, 3, 1, 27, 1, 3, 29, 3, 1, 29, 1, 3, 31, 3

Offset

1,2

References

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 601.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

G. Xiao, Contfrac

Index entries for continued fractions for constants

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

Formula

a(1)=1, a(2)=2, a(3)=1, a(4)=3, a(5)=1, a(6)=1, a(7)=1, a(8)=3, then for k>=1 a(6*k+3)=a(6*k+6)=2*k+1, a(6*k+4)=a(6*k+8)=3, a(6*k+5)=a(6*k+7)=1. - Benoit Cloitre, Apr 08 2003

From Colin Barker, May 16 2016: (Start)

a(n) = a(n-3)+a(n-6)-a(n-9) for n>9.

G.f.: x*(1+2*x+x^2+2*x^3-x^4-3*x^6+x^8-x^10) / ((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)^2).

(End)

Example

1.359140914229522617680143735... = 1 + 1/(2 + 1/(1 + 1/(3 + 1/(1 + ...)))). - Harry J. Smith, May 10 2009

Mathematica

ContinuedFraction[E/2, 94] (* Jean-François Alcover, Apr 01 2011 *)
Join[{1, 2}, LinearRecurrence[{0, 0, 1, 0, 0, 1, 0, 0, -1}, {1, 3, 1, 1, 1, 3, 3, 3, 1}, 92]] (* Ray Chandler, Sep 03 2015 *)

Prog

(PARI) { allocatemem(932245000); default(realprecision, 55000); x=contfrac(exp(1)/2); for (n=1, 20000, write("b006083.txt", n, " ", x[n])); } \\ Harry J. Smith, May 10 2009
(PARI) Vec(x*(1+2*x+x^2+2*x^3-x^4-3*x^6+x^8-x^10) / ((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)^2) + O(x^50)) \\ Colin Barker, May 16 2016

Crossrefs

Cf. A019739 = Decimal expansion. - Harry J. Smith, May 10 2009

Sequence in context: A362681 A320077 A325522 * A338713 A080301 A326674

Adjacent sequences: A006080 A006081 A006082 * A006084 A006085 A006086

Keyword

cofr,nonn,easy

Author

N. J. A. Sloane, Jeffrey Shallit

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

Status

approved