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A006084 Continued fraction for e/3.
(Formerly M4591)
4
0, 1, 9, 1, 1, 1, 5, 1, 1, 1, 1, 26, 1, 1, 3, 5, 1, 3, 1, 1, 44, 1, 1, 5, 5, 1, 5, 1, 1, 62, 1, 1, 7, 5, 1, 7, 1, 1, 80, 1, 1, 9, 5, 1, 9, 1, 1, 98, 1, 1, 11, 5, 1, 11, 1, 1, 116, 1, 1, 13, 5, 1, 13, 1, 1, 134, 1, 1, 15, 5, 1, 15, 1, 1, 152, 1, 1, 17, 5, 1, 17, 1, 1, 170, 1, 1, 19, 5, 1, 19, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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 (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).

FORMULA

First eight terms are 0, 1, 9, 1, 1, 1, 5, 1; then a(9k)=2k-1, a(9k+1)=1, a(9k+2)=1, a(9k+3)=18k+8, a(9k+4)=1, a(9k+5)=1, a(9k+6)=2k+1, a(9k+7)=5, a(9k+8)=1. - Benoit Cloitre, Apr 08 2003

G.f.: x^2*(1+9*x+x^2+x^3+x^4+5*x^5+x^6+x^7+x^8-x^9+8*x^10-x^11-x^12+x^13-5*x^14-x^15+x^16-x^17+x^19) / ((1-x)^2*(1+x+x^2)^2*(1+x^3+x^6)^2). - Colin Barker, May 16 2016

EXAMPLE

0.906093942819681745120095823... = 0 + 1/(1 + 1/(9 + 1/(1 + 1/(1 + ...)))). - Harry J. Smith, May 12 2009

MATHEMATICA

ContinuedFraction[E/3, 100] (* Harvey P. Dale, Oct 14 2013 *)

Join[{0, 1, 9}, LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1}, {1, 1, 1, 5, 1, 1, 1, 1, 26, 1, 1, 3, 5, 1, 3, 1, 1, 44}, 89]] (* Ray Chandler, Sep 03 2015 *)

PROG

(PARI) { allocatemem(932245000); default(realprecision, 54000); x=contfrac(exp(1)/3); for (n=1, 20000, write("b006084.txt", n, " ", x[n])); } \\ Harry J. Smith, May 12 2009

(PARI) concat(0, Vec(x^2*(1+9*x+x^2+x^3+x^4+5*x^5+x^6+x^7+x^8-x^9+8*x^10-x^11-x^12+x^13-5*x^14-x^15+x^16-x^17+x^19)/((1-x)^2*(1+x+x^2)^2*(1+x^3+x^6)^2) + O(x^50))) \\ Colin Barker, May 16 2016

CROSSREFS

Cf. A019740 = Decimal expansion. - Harry J. Smith, May 12 2009

Sequence in context: A087968 A110483 A010164 * A059928 A293724 A283989

Adjacent sequences:  A006081 A006082 A006083 * A006085 A006086 A006087

KEYWORD

nonn,cofr,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

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Last modified January 22 18:48 EST 2019. Contains 319365 sequences. (Running on oeis4.)