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A002531
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Numerators of continued fraction convergents to sqrt(3).
(Formerly M1340 N0513)
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16
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1, 1, 2, 5, 7, 19, 26, 71, 97, 265, 362, 989, 1351, 3691, 5042, 13775, 18817, 51409, 70226, 191861, 262087, 716035, 978122, 2672279, 3650401, 9973081, 13623482, 37220045, 50843527, 138907099, 189750626, 518408351, 708158977, 1934726305
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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
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
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REFERENCES
| 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
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.
Index entries for "core" sequences
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FORMULA
| G.f.: (1+x-2*x^2+x^3)/(1-4*x^2+x^4).
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.
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
a(n) = ((1+sqrt(3))^n+(1-sqrt(3))^n)/(2*2^floor(n/2)). - Bruno Berselli, Nov 10 2011
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EXAMPLE
| 1+1/(1+1/(2+1/(1+1/2)))=19/11 so a(5)=19.
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
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MAPLE
| 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) ];
with(numtheory): tp := cfrac (tan(Pi/3), 100): seq(nthnumer(tp, i), i=-1..32 ); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2007
A002531:=(1+z-2*z**2+z**3)/(1-4*z**2+z**4); [From S. Plouffe, see his 1992 dissertation.]
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MATHEMATICA
| Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[3], n]]], {n, 1, 40}], 1, 1] (* Stefan Steinerberger, Apr 01 2006 *)
Join[{1}, Numerator[Convergents[Sqrt[3], 40]]] (* From Harvey P. Dale, Jan 23 2012 *)
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PROG
| (PARI) a(n)=if(n<0, 0, contfracpnqn(vector(n, i, 1+(i>1)*(i%2)))[1, 1])
(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]
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CROSSREFS
| Bisections are A001075 and A001834.
Cf. A002530, A048788.
Cf. A002316.
Cf. A083332, A199710.
Sequence in context: A174362 A042809 A108413 * A042449 A046115 A089443
Adjacent sequences: A002528 A002529 A002530 * A002532 A002533 A002534
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KEYWORD
| nonn,frac,easy,core,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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