A006085 Continued fraction for e/4.

0, 1, 2, 8, 3, 1, 1, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 7, 1, 2, 2, 1, 1, 1, 3, 7, 1, 3, 2, 1, 1, 1, 4, 7, 1, 4, 2, 1, 1, 1, 5, 7, 1, 5, 2, 1, 1, 1, 6, 7, 1, 6, 2, 1, 1, 1, 7, 7, 1, 7, 2, 1, 1, 1, 8, 7, 1, 8, 2, 1, 1, 1, 9, 7, 1, 9, 2, 1, 1, 1, 10, 7, 1, 10, 2, 1, 1, 1, 11, 7, 1, 11, 2, 1, 1, 1, 12, 7, 1, 12, 2, 1

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy

Examples

			0.679570457114761308840071867... = 0 + 1/(1 + 1/(2 + 1/(8 + 1/(3 + ...)))). - _Harry J. Smith_, May 10 2009
		

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 (1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1).

Crossrefs

Cf. A019741 = Decimal expansion. - Harry J. Smith, May 10 2009

Programs

Mathematica

ContinuedFraction[E/4,120]  (* Harvey P. Dale, Apr 01 2011 *)
Join[{0, 1, 2, 8, 3},LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1},{1, 1, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 7, 1, 2},97]] (* Ray Chandler, Sep 03 2015 *)

PARI

{ allocatemem(932245000); default(realprecision, 40000); x=contfrac(exp(1)/4); for (n=1, 20000, write("b006085.txt", n, " ", x[n])); } \\ Harry J. Smith, May 10 2009

PARI

concat(0, Vec(x*(1+x+7*x^2-4*x^3+5*x^4-4*x^5+5*x^6-4*x^7+9*x^8-12*x^9-3*x^10-x^11-x^13-6*x^16+7*x^17+x^18)/((1-x)^2*(1+x)*(1+x^2)^2*(1+x^4)^2) + O(x^50))) \\ Colin Barker, May 16 2016

Formula

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

G.f.: x*(1+x+7*x^2-4*x^3+5*x^4-4*x^5+5*x^6-4*x^7+9*x^8-12*x^9-3*x^10-x^11-x^13-6*x^16+7*x^17+x^18) / ((1-x)^2*(1+x)*(1+x^2)^2*(1+x^4)^2). - Colin Barker, May 16 2016

Extensions

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