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A005667 Numerators of continued fraction convergents to sqrt(10).
(Formerly M3056)
10
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.

Tanya Khovanova, Recursive Sequences

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 sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (6,1).

FORMULA

a(n) = 6a(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.

Binomial transform of A084132. E.g.f. : exp(3x)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, 2k)10^k3^(n-2k)}. - Paul Barry, Nov 15 2003

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, 50}]] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)

CoefficientList[Series[(1 - 3 x) / (1 - 6 x - x^2), {x, 0, 50}], 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 *)

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

CROSSREFS

Cf. A084134, A005668.

Sequence in context: A037781 A037585 A084133 * A098444 A290477 A221184

Adjacent sequences:  A005664 A005665 A005666 * A005668 A005669 A005670

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

Chebyshev comments from Wolfdieter Lang, Jan 10 2003

STATUS

approved

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Last modified September 24 22:31 EDT 2017. Contains 292441 sequences.