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 A002531 Numerators of continued fraction convergents to sqrt(3). (Formerly M1340 N0513) 16

%I M1340 N0513

%S 1,1,2,5,7,19,26,71,97,265,362,989,1351,3691,5042,13775,18817,51409,

%T 70226,191861,262087,716035,978122,2672279,3650401,9973081,13623482,

%U 37220045,50843527,138907099,189750626,518408351,708158977,1934726305

%N Numerators of continued fraction convergents to sqrt(3).

%C 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 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 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 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 181.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D 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 Harry J. Smith, <a href="/A002531/b002531.txt">Table of n, a(n) for n=0,...,2000</a>

%H _Simon 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 _Simon 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 <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F G.f.: (1+x-2*x^2+x^3)/(1-4*x^2+x^4).

%F a(2*n) = a(2*n-1) + a(2*n-2), a(2*n+1) = 2*a(2*n) + a(2*n-1), n>0.

%F a(2*n) = (1/2)*((2+sqrt(3))^n+(2-sqrt(3))^n); a(2*n) = A003500(n)/2; a(2*n+1) = round( 1/(1+sqrt(3))*(2+sqrt(3))^n) - Benoit Cloitre, Dec 15 2002

%F a(n) = ((1+sqrt(3))^n+(1-sqrt(3))^n)/(2*2^floor(n/2)). - Bruno Berselli, Nov 10 2011

%F a(2*n) = (-1)^n*T(2*n,u) and a(2*n+1) = (-1)^n*1/u*T(2*n+1,u), where u = sqrt(-1/2) and T(n,x) denotes the Chebyshev polynomial of the first kind. - Peter Bala, May 01 2012

%e 1+1/(1+1/(2+1/(1+1/2)))=19/11 so a(5)=19.

%e 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 := 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 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:=(1+z-2*z**2+z**3)/(1-4*z**2+z**4); [From _Simon Plouffe_, see his 1992 dissertation.]

%t Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[3], n]]], {n, 1, 40}], 1, 1] (* Stefan Steinerberger, Apr 01 2006 *)

%t Join[{1},Numerator[Convergents[Sqrt[3],40]]] (* From Harvey P. Dale, Jan 23 2012 *)

%o (PARI) a(n)=if(n<0,0,contfracpnqn(vector(n,i,1+(i>1)*(i%2)))[1,1])

%o (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, Jun 01 2009]

%Y Bisections are A001075 and A001834.

%Y Cf. A002530, A048788.

%Y Cf. A002316.

%Y Cf. A083332, A199710.

%K nonn,frac,easy,core,nice

%O 0,3

%A _N. J. A. Sloane_.

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Last modified June 20 05:50 EDT 2013. Contains 226419 sequences.