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A002531 Numerators of continued fraction convergents to sqrt(3).
(Formerly M1340 N0513)
21
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; text; internal format)
OFFSET

0,3

COMMENTS

For the denominators see A002530.

Consider the mapping f(a/b) = (a + 3*b)/(a + b). Taking a = b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the convergents 1/1, 2/1, 5/3, 7/4, 19/11, ... converging to 3^(1/2). Sequence contains the numerators. - Amarnath Murthy, 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, Jun 11 2007

2^(-floor(n/2))*(1 + sqrt(3))^n = a(n) + A002530(n)*sqrt(3); integers in the real quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 10 2018

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.

LINKS

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]

Hein van Winkel, Q-quadrangles inscribed in a circle, 2014. See Table 1. [Reference from Antreas Hatzipolakis, Jul 14 2014]

Index entries for "core" sequences

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Index entries for sequences related to Chebyshev polynomials.

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

a(n) = A080040(n)/(2*2^floor(n/2)). - Ralf Stephan, Sep 08 2013

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

a(n) = (-sqrt(2)*i)^n*T(n, sqrt(2)*i/2)*2^(-floor(n/2)) = A026150(n)*2^(-floor(n/2)), n >= 0, with i = sqrt(-1) and the Chebyshev T polynomials (A053120). - Wolfdieter Lang, Feb 10 2018

From Franck Maminirina Ramaharo, Nov 14 2018: (Start)

a(n) = ((1 - sqrt(2))*(-1)^n + 1 + sqrt(2))*(((sqrt(2) - sqrt(6))/2)^n + ((sqrt(6) + sqrt(2))/2)^n)/4.

E.g.f.: cosh(sqrt(3/2)*x)*(sqrt(2)*sinh(x/sqrt(2)) + cosh(x/sqrt(2))). (End)

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.

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, Feb 07 2007

A002531:=(1+z-2*z**2+z**3)/(1-4*z**2+z**4); # Simon Plouffe; see his 1992 dissertation

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]]] (* Harvey P. Dale, Jan 23 2012 *)

CoefficientList[Series[(1 + x - 2 x^2 + x^3)/(1 - 4 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 01 2014 *)

LinearRecurrence[{0, 4, 0, -1}, {1, 1, 2, 5}, 35] (* Robert G. Wilson v, Feb 11 2018 *)

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); ); } \\ Harry J. Smith, Jun 01 2009

(MAGMA) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1 +x-2*x^2+x^3)/(1-4*x^2+x^4))); // G. C. Greubel, Nov 16 2018

(Sage) s=((1+x-2*x^2+x^3)/(1-4*x^2+x^4)).series(x, 40); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 16 2018

(GAP) a:=[1, 1, 2, 5];; for n in [5..40] do a[n]:=4*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Nov 16 2018

CROSSREFS

Bisections are A001075 and A001834.

Cf. A002530 (denominators), A048788.

Cf. A002316.

Cf. A083332, A199710, A026150, A053120.

Sequence in context: A128005 A045359 A042809 * A108413 A042449 A046115

Adjacent sequences:  A002528 A002529 A002530 * A002532 A002533 A002534

KEYWORD

nonn,frac,easy,core,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 18 13:50 EDT 2019. Contains 326100 sequences. (Running on oeis4.)