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 A005667 Numerators of continued fraction convergents to sqrt(10). (Formerly M3056) 19
 1, 3, 19, 117, 721, 4443, 27379, 168717, 1039681, 6406803, 39480499, 243289797, 1499219281, 9238605483, 56930852179, 350823718557, 2161873163521, 13322062699683, 82094249361619, 505887558869397, 3117419602578001, 19210405174337403, 118379850648602419 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(2*n+1) with b(2*n+1) := A005668(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = -1, a(2*n) with b(2*n) := A005668(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = +1 (cf. Emerson reference). Bisection: a(2*n) = T(n,19) = A078986(n), n >= 0 and a(2*n+1) = 3*S(2*n, 2*sqrt(10)), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. See A053120, resp. A049310. REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Thm. 1, p. 233. R. K. Guy, Letter to N. J. A. Sloane, 1987 Tanya Khovanova, Recursive Sequences Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019. 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. Index entries for linear recurrences with constant coefficients, signature (6,1). FORMULA a(n) = 6*a(n-1) + a(n-2). G.f.: (1-3*x)/(1-6*x-x^2). a(n) = ((-i)^n)*T(n, 3*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. From Paul Barry, Nov 15 2003: (Start) Binomial transform of A084132. E.g.f.: exp(3*x)*cosh(sqrt(10)*x). a(n) = ((3+sqrt(10))^n + (3-sqrt(10))^n)/2. a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k) * 10^k * 3^(n-2*k). (End) a(n) = (-1)^n * a(-n) for all n in Z. - Michael Somos, Jul 14 2018 a(n) = Lucas(n,6)/2, Lucas polynomial, L(n,x), evaluated at x = 6. - G. C. Greubel, Jun 06 2019 EXAMPLE G.f. = 1 + 3*x + 19*x^2 + 117*x^3 + 721*x^4 + 4443*x^5 + 27379*x^6 + ... - Michael Somos, Jul 14 2018 MAPLE A005667:=(-1+3*z)/(-1+6*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation MATHEMATICA Join[{1}, Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[10], n]]], {n, 1, 30}]] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *) CoefficientList[Series[(1-3x)/(1-6x-x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 09 2013 *) Join[{1}, Numerator[Convergents[Sqrt[10], 30]]] (* or *) LinearRecurrence[ {6, 1}, {1, 3}, 30] (* Harvey P. Dale, Aug 22 2016 *) a[ n_] := (-I)^n ChebyshevT[ n, 3 I]; (* Michael Somos, Jul 14 2018 *) LucasL[Range[0, 30], 6]/2 (* G. C. Greubel, Jun 06 2019 *) PROG (MAGMA) I:=[1, 3]; [n le 2 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 09 2013 (PARI) a(n)=([0, 1; 1, 6]^n*[1; 3])[1, 1] \\ Charles R Greathouse IV, Jun 11 2015 (Sage) ((1-3*x)/(1-6*x-x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 06 2019 CROSSREFS Cf. A010467, A040006, A084134, A005668 (denominators). Sequence in context: A037781 A037585 A084133 * A098444 A290477 A321002 Adjacent sequences:  A005664 A005665 A005666 * A005668 A005669 A005670 KEYWORD nonn,frac,easy AUTHOR EXTENSIONS Chebyshev comments from Wolfdieter Lang, Jan 10 2003 STATUS approved

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Last modified October 14 02:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)