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%I M1822 #29 May 17 2016 04:04:23
%S 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,
%T 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,
%U 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
%N Continued fraction for e/4.
%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 601.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Harry J. Smith, <a href="/A006085/b006085.txt">Table of n, a(n) for n = 1..20000</a>
%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1).
%F 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
%F 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
%e 0.679570457114761308840071867... = 0 + 1/(1 + 1/(2 + 1/(8 + 1/(3 + ...)))). - _Harry J. Smith_, May 10 2009
%t ContinuedFraction[E/4,120] (* _Harvey P. Dale_, Apr 01 2011 *)
%t 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 *)
%o (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
%o (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
%Y Cf. A019741 = Decimal expansion. - _Harry J. Smith_, May 10 2009
%K nonn,cofr,easy
%O 1,3
%A _N. J. A. Sloane_
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003
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