%I A005246 M0829
%S A005246 1,1,1,2,3,7,11,26,41,97,153,362,571,1351,2131,5042,7953,18817,29681,
%T A005246 70226,110771,262087,413403,978122,1542841,3650401,5757961,13623482,
%U A005246 21489003,50843527,80198051,189750626,299303201,708158977,1117014753
%N A005246 a(n)=(1+a(n-1)a(n-2))/a(n-3).
%C A005246 For n >= 4 we have the linear recurrence a(n) = 4*a(n-2) - a(n-4). - 
               Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 04 2001
%C A005246 Integer solutions to the equation floor(sqrt(3)*x^2)=x*floor(sqrt(3)*x) 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 18 2004
%C A005246 For n>2, a(n) is the smallest integer > a(n-1) such that sqrt(3)*a(n) 
               is closer to and greater than an integer than sqrt(3)*a(n-1). i.e. 
               a(n) is the smallest integer > a(n-1) such that frac(sqrt(3)*a(n))<frac(sqrt(3)*a(n-1)). 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 20 2003
%C A005246 The lower principal and intermediate convergents to 3^(1/2), beginning 
               with 1/1, 3/2, 5/3, 12/7, 19/11, form a strictly increasing sequence; 
               essentially, numerators=A143643 and denominators=A005246. - Clark 
               Kimberling (ck6(AT)evansville.edu), Aug 27 2008
%D A005246 T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 
               263-271.
%D A005246 Clark Kimberling, "Best lower and upper approximates to irrational numbers,
               " Elemente der Mathematik, 52 (1997) 122-126.
%D A005246 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, 
               New York, 1966.
%D A005246 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005246 T. D. Noe, <a href="b005246.txt">Table of n, a(n) for n=0..500</a>
%H A005246 Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, <a href="http:/
               /www.cs.uwaterloo.ca/journals/JIS/index.html">A Note on Rational 
               Succession Rules</a>, J. Integer Seqs., Vol. 6, 2003.
%H A005246 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005246 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A005246 G.f.: (1+x-3*x^2-2*x^3)/(1-4*x^2+x^4).
%F A005246 lim n ->infinity a(2n+1)/a(2n) = (3+sqrt(3))/3 =1.5773502... lim n ->
               infinity a(2n)/a(2n-1) = (3+sqrt(3))/2 = 2.3660254.... - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Aug 07 2002
%F A005246 A101265(n) = a(n)*a(n+1). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Apr 24 2006
%F A005246 a(2-n)=a(n). - Michael Somos Nov 15 2006
%F A005246 For n>2: a(n) = a(n-1) + SUM(a(2*k): 1 <= k < n/2). - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Dec 16 2007
%p A005246 A005246:=-(-1-z+2*z**2+z**3)/(1-4*z**2+z**4); [Conjectured by S. Plouffe 
               in his 1992 dissertation. Gives sequence except for one of the leading 
               1's.]
%o A005246 (PARI) {a(n)=if(n<0, n=2-n); polcoeff((1+x-3*x^2-2*x^3)/(1-4*x^2+x^4)+x*O(x^n), 
               n)} /* Michael Somos Nov 15 2006 */
%Y A005246 Bisections are A001835 and A001075.
%Y A005246 Cf. A101265.
%Y A005246 Sequence in context: A007481 A121268 A101173 this_sequence A116406 A112843 
               A036651
%Y A005246 Adjacent sequences: A005243 A005244 A005245 this_sequence A005247 A005248 
               A005249
%K A005246 easy,nonn,nice
%O A005246 0,4
%A A005246 N. J. A. Sloane (njas(AT)research.att.com).
%E A005246 More terms from Michael Somos, Aug 01 2001