0,4

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, ...

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

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

Comment from Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 (Start): Related convergents (numerator/denominator):

lower principal convergents: A001834/A001835

upper principal convergents: A001075/A001353

intermediate convergents: A005320/A001075

principal and intermediate convergents: A143642/A140827

lower principal and intermediate convergents: A143643/A005246 (End)

Row sums of triangle A152063 = (1, 3, 4, 11,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]

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.

Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126.

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Russell Lyons, A birds-eye view of uniform spanning trees and forests, in Microsurveys in Discrete Probability, AMS, 1998.

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

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

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

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.

Harry J. Smith, Table of n, a(n) for n=0,...,2000

Mario Catalani, Sequences related to convergents to square root of rationals

C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

R. Walsmith, DCL-Chemy Transforms Fibonacci-type Sequences to Arrays page 3.

Index entries for "core" sequences

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

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

G.f.: x(1+x-x^2)/(1-4x^2+x^4). a(n) = 4a(n-2)+a(n-4) = -(-1)^n a(-n).

a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 2a(2n) + a(2n-1).

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

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

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

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

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) ];

with(numtheory): tng := cfrac (tan(Pi/3), 100): seq(nthdenom (tng, i), i=-1..32 ); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2007

A002530:=-(-1-z+z**2)/(1-4*z**2+z**4); [Conjectured (correctly) by S. Plouffe in his 1992 dissertation.]

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[3], n]]], {n, 1, 40}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006

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

(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]

Cf. A002531 (numerators), A048788, A003297.

Bisections: A001353 and A001835.

Cf. A152063 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]

Sequence in context: A085368 A041405 A042483 this_sequence A042709 A042327 A046114

Adjacent sequences: A002527 A002528 A002529 this_sequence A002531 A002532 A002533

nonn,easy,frac,core,nice

N. J. A. Sloane (njas(AT)research.att.com).