%I A002530 M2363 N0934
%S A002530 0,1,1,3,4,11,15,41,56,153,209,571,780,2131,2911,7953,10864,29681,
%T A002530 40545,110771,151316,413403,564719,1542841,2107560,5757961,7865521,
%U A002530 21489003,29354524,80198051,109552575,299303201,408855776,1117014753
%N A002530 Denominators of continued fraction convergents to sqrt(3).
%C A002530 Also denominators of continued fraction convergents to sqrt(3) - 1. See 
               A048788 for numerators. - N. J. A. Sloane (njas(AT)research.att.com), 
               Dec 17 2007. Convergents are 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/
               56, 112/153, ...
%C A002530 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 denominators. The 
               same mapping for N i.e. f(a/b) = (a + Nb)/(a+b) gives fractions converging 
               to N^(1/2). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 
               22 2003
%C A002530 Sqrt(3) = 2/2 + 2/3 + 2/(3*11) + 2/(11*41) + 2/(41*153) + 2/(153*571),
               ...; where the sum of the first 6 terms of this series = 1.7320490367... 
               and sqrt(3) = 1.7320508075... - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 15 2007
%C A002530 Comment from Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 (Start): 
               Related convergents (numerator/denominator):
%C A002530 lower principal convergents: A001834/A001835
%C A002530 upper principal convergents: A001075/A001353
%C A002530 intermediate convergents: A005320/A001075
%C A002530 principal and intermediate convergents: A143642/A140827
%C A002530 lower principal and intermediate convergents: A143643/A005246 (End)
%C A002530 Row sums of triangle A152063 = (1, 3, 4, 11,...). [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Nov 26 2008]
%D A002530 Fraenkel, Aviezri S.; Levitt, Jonathan; Shimshoni, Michael; Characterization 
               of the set of values f(n)=[n alpha], n=1,2,... Discrete Math. 2 (1972), 
               no.4, 335-345.
%D A002530 Clark Kimberling, "Best lower and upper approximates to irrational numbers,
               " Elemente der Mathematik, 52 (1997) 122-126.
%D A002530 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, 
               New York, 1966.
%D A002530 Russell Lyons, A birds-eye view of uniform spanning trees and forests, 
               in Microsurveys in Discrete Probability, AMS, 1998.
%D A002530 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 
               2nd ed., Wiley, NY, 1966, p. 181.
%D A002530 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002530 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002530 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 A002530 Harry J. Smith, <a href="b002530.txt">Table of n, a(n) for n=0,...,2000</
               a>
%H A002530 Mario Catalani, <a href="http://arXiv.org/abs/math.NT/0305270">Sequences 
               related to convergents to square root of rationals</a>
%H A002530 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., 
               Vol. 6, 2003.
%H A002530 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 A002530 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 A002530 R. Walsmith, <a href="http://fibonacci-arrays.com/fibars.pdf">DCL-Chemy 
               Transforms Fibonacci-type Sequences to Arrays</a> page 3.
%H A002530 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A002530 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A002530 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A002530 a(2n) = ((2+sqrt(3))^n-(2-sqrt(3))^n)/(2*sqrt(3)); a(2n) = A001353(n); 
               a(2n-1) = ceil((1+1/sqrt(3))/2*(2+sqrt(3))^n) = ((3+sqrt(3))^(2n-1)+(3-sqrt(3))^(2n-1))/
               6^n; a(2n-1) = A001835(n).- Benoit Cloitre, Dec 15 2002
%F A002530 G.f.: x(1+x-x^2)/(1-4x^2+x^4). a(n) = 4a(n-2)+a(n-4) = -(-1)^n a(-n).
%F A002530 a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 2a(2n) + a(2n-1).
%F A002530 a(n+1) = sum{k=0..floor(n/2), binomial(n-k, k)2^floor((n-2k)/2) } - Paul 
               Barry (pbarry(AT)wit.ie), Jul 13 2004
%F A002530 a(n) = sum_{k=0..floor(n/2)} binomial(floor(n/2)+k,floor((n-1)/2-k))*2^k). 
               - Paul Barry (pbarry(AT)wit.ie), Jun 22 2005
%e A002530 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
%e A002530 1+1/(1+1/(2+1/(1+1/2))))=19/11 so a(5)=11.
%p A002530 a := proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 
               then 1 elif n=3 then 3 else 4*a(n-2)-a(n-4) fi end; [ seq(a(i),i=0..50) 
               ];
%p A002530 with(numtheory): tng := cfrac (tan(Pi/3),100): seq(nthdenom (tng,i), 
               i=-1..32 ); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 
               07 2007
%p A002530 A002530:=-(-1-z+z**2)/(1-4*z**2+z**4); [Conjectured (correctly) by S. 
               Plouffe in his 1992 dissertation.]
%t A002530 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[3],n]]], 
               {n,1,40}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 01 2006
%o A002530 (PARI) a(n)=if(n<0,-(-1)^n*a(-n),contfracpnqn(vector(n,i,1+(i>1)*(i%2)))[2,
               1])
%o A002530 (PARI) { default(realprecision, 2000); for (n=0, 2000, a=contfracpnqn(vector(n, 
               i, 1+(i>1)*(i%2)))[2, 1]; write("b002530.txt", n, " ", a); ); } [From 
               Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 01 2009]
%Y A002530 Cf. A002531 (numerators), A048788, A003297.
%Y A002530 Bisections: A001353 and A001835.
%Y A002530 Cf. A152063 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]
%Y A002530 Sequence in context: A085368 A041405 A042483 this_sequence A042709 A042327 
               A046114
%Y A002530 Adjacent sequences: A002527 A002528 A002529 this_sequence A002531 A002532 
               A002533
%K A002530 nonn,easy,frac,core,nice
%O A002530 0,4
%A A002530 N. J. A. Sloane (njas(AT)research.att.com).