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 A002531 a(2*n) = a(2*n-1) + a(2*n-2), a(2*n+1) = 2*a(2*n) + a(2*n-1); a(0) = a(1) = 1. (Formerly M1340 N0513) 28
 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 Numerators of continued fraction convergents to sqrt(3), for n >= 1. 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 MacTutor, D'Arcy Thompson on Greek irrationals Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy] D'Arcy Thompson, Excess and Defect: Or the Little More and the Little Less, Mind, New Series, Vol. 38, No. 149 (Jan., 1929), pp. 43-55 (13 pages). See page 48. Hein van Winkel, Q-quadrangles inscribed in a circle, 2014. See Table 1. [Reference from Antreas Hatzipolakis, Jul 14 2014] Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1). 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) a(n) = (-1)^n*a(-n) for all n in Z. - Michael Somos, Mar 22 2022 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. G.f. = 1 + x + 2*x^2 + 5*x^3 + 7*x^4 + 19*x^5 + 26*x^6 + 71*x^7 + ... - Michael Somos, Mar 22 2022 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 *) a[ n_] := ChebyshevT[n, Sqrt[-1/2]]*Sqrt[2]^Mod[n, 2]/I^n //Simplify; (* Michael Somos, Mar 22 2022 *) PROG (PARI) a(n)=contfracpnqn(vector(n, i, 1+(i>1)*(i%2)))[1, 1] (PARI) apply( {A002531(n, w=quadgen(12))=real((2+w)^(n\/2)*if(bittest(n, 0), w-1, 1))}, [0..30]) \\ M. F. Hasler, Nov 04 2019 (MAGMA) m:=40; R:=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 EXTENSIONS Name edited (as by discussion in A002530) by M. F. Hasler, Nov 04 2019 STATUS approved

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Last modified August 12 15:02 EDT 2022. Contains 356077 sequences. (Running on oeis4.)