%I A002531 M1340 N0513
%S A002531 1,1,2,5,7,19,26,71,97,265,362,989,1351,3691,5042,13775,18817,51409,
%T A002531 70226,191861,262087,716035,978122,2672279,3650401,9973081,13623482,
%U A002531 37220045,50843527,138907099,189750626,518408351,708158977,1934726305
%N A002531 Numerators of continued fraction convergents to sqrt(3).
%C A002531 Consider the mapping f(a/b) = (a + 3b)/(a + b). Taking a = b = 1 to start 
               with and carrying out this mapping repeatedly on each new (reduced) 
               rational number gives the following sequence 1/1,2/1,5/3,7/4,19/11,
               ... converging to 3^(1/2). Sequence contains the numerators. - Amarnath 
               Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
%C A002531 In the Murthy comment if we take a=0, b=1 then the denominator of the 
               reduced fraction is a(n+1). A083336(n)/a(n+1) converges to sqrt(3). 
               - Mario Catalani (mario.catalani(AT)unito.it), Apr 26 2003
%C A002531 If signs are disregarded, all terms of A002316 appear to be elements 
               of this sequence. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 
               Jun 11 2007
%D A002531 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 
               2nd ed., Wiley, NY, 1966, p. 181.
%D A002531 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002531 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002531 A. Tarn, Approximations to certain square roots and the series of numbers 
               connected therewith, Mathematical Questions and Solutions from the 
               Educational Times, 1 (1916), 8-12.
%H A002531 Harry J. Smith, <a href="b002531.txt">Table of n, a(n) for n=0,...,2000</
               a>
%H A002531 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 A002531 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.
%H A002531 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A002531 G.f.: (1+x-2x^2+x^3)/(1-4x^2+x^4).
%F A002531 a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 2a(2n) + a(2n-1), n>0.
%F A002531 a(2n)=(1/2)*((2+sqrt(3))^n+(2-sqrt(3))^n); a(2n)=A003500(n)/2; a(2n+1) 
               = round( 1/(1+sqrt(3))*(2+sqrt(3))^n) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Dec 15 2002
%e A002531 1+1/(1+1/(2+1/(1+1/2)))=19/11 so a(5)=19.
%e A002531 Convergents are 1, 2, 5/3, 7/4, 19/11, 26/15, 71/41, 97/56, 265/153, 
               362/209, 989/571, 1351/780, 3691/2131, ... = A002531/A002530
%p A002531 A002531 := proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif 
               n=2 then 1 elif type(n,odd) then A002531(n-1)+A002531(n-2) else 2*A002531(n-1)+A002531(n-2) 
               fi; end; [ seq(A002531(n), n=0..50) ];
%p A002531 with(numtheory): tp := cfrac (tan(Pi/3),100): seq(nthnumer(tp,i), i=-1..32 
               ); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2007
%p A002531 A002531:=(1+z-2*z**2+z**3)/(1-4*z**2+z**4); [Conjectured (correctly) 
               by S. Plouffe in his 1992 dissertation.]
%t A002531 Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[3], 
               n]]], {n, 1, 40}], 1, 1] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 01 2006
%o A002531 (PARI) a(n)=if(n<0,0,contfracpnqn(vector(n,i,1+(i>1)*(i%2)))[1,1])
%o A002531 (PARI) { default(realprecision, 2000); for (n=0, 2000, a=contfracpnqn(vector(n, 
               i, 1+(i>1)*(i%2)))[1, 1]; write("b002531.txt", n, " ", a); ); } [From 
               Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 01 2009]
%Y A002531 Bisections are A001075 and A001834.
%Y A002531 Cf. A002530, A048788.
%Y A002531 Cf. A002316.
%Y A002531 Sequence in context: A045359 A042809 A108413 this_sequence A042449 A046115 
               A089443
%Y A002531 Adjacent sequences: A002528 A002529 A002530 this_sequence A002532 A002533 
               A002534
%K A002531 nonn,frac,easy,core,nice
%O A002531 0,3
%A A002531 N. J. A. Sloane (njas(AT)research.att.com).