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A002530
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Denominators of continued fraction convergents to sqrt(3).
(Formerly M2363 N0934)
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25
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0, 1, 1, 3, 4, 11, 15, 41, 56, 153, 209, 571, 780, 2131, 2911, 7953, 10864, 29681, 40545, 110771, 151316, 413403, 564719, 1542841, 2107560, 5757961, 7865521, 21489003, 29354524, 80198051, 109552575, 299303201, 408855776, 1117014753
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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]
Also number of domino tilings of the 3 X (n-1) rectangle with upper left corner removed iff n is even. For n=4 the 4 domino tilings of the 3 X 3 rectangle with upper left corner removed are:
. .___. . .___. . .___. . .___.
._|___| ._|___| ._| | | ._|___|
| |___| | | | | | |_|_| |___| |
|_|___| |_|_|_| |_|___| |___|_| - Alois P. Heinz, Apr 13 2011
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REFERENCES
| Marcia Edson, Scott Lewis and Omer Yayenie, THE K-PERIODIC FIBONACCI SEQUENCE AND AN EXTENDED BINET'S FORMULA, INTEGERS 11 (2011) #A32.
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.
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LINKS
| 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
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FORMULA
| 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
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EXAMPLE
| 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.
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MAPLE
| 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.]
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MATHEMATICA
| Join[{0}, Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[3], n]]], {n, 1, 40}]] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
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PROG
| (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]
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CROSSREFS
| 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 * A042709 A042327 A046114
Adjacent sequences: A002527 A002528 A002529 * A002531 A002532 A002533
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KEYWORD
| nonn,easy,frac,core,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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