A041024 Numerators of continued fraction convergents to sqrt(17).

4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236

Offset

0,1

Comments

a(2*n+1) with b(2*n+1) := A041025(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 17*b^2 = +1, a(2*n) with b(2*n) := A041025(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 17*b^2 = -1 (cf. Emerson reference).

Bisection: a(2*n) = 4*S(2*n,2*sqrt(17)) = 4*A078989(n), n >= 0 and a(2*n+1) = T(n+1,33), n >= 0, with S(n,x), resp. T(n,x), Chebyshev's polynomials of the second, resp. first kind. See A049310, resp. A053120. - Wolfdieter Lang, Jan 10 2003

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (8,1).

Formula

G.f.: (4+x)/(1-8*x-x^2).

a(n) = 4*A041025(n) + A041025(n-1).

a(n) = ((-i)^(n+1))*T(n+1, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = -1.

a(n) = 8*a(n-1) + a(n-2), n > 1. - Philippe Deléham, Nov 20 2008

a(n) = ((4 + sqrt(17))^n + (4 - sqrt(17))^n)/2. - Sture Sjöstedt, Dec 08 2011

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[17], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011*)
LinearRecurrence[{8, 1}, {4, 33}, 25] (* Sture Sjöstedt, Dec 07 2011 *)
CoefficientList[Series[(4 + x)/(1 - 8 x - x^2), {x, 0, 30}], x]  (* Vincenzo Librandi_, Oct 28 2013 *)

Crossrefs

Cf. A010473, A040012, A041025.

Sequence in context: A081007 A213168 A203212 * A088317 A257068 A246806

Adjacent sequences: A041021 A041022 A041023 * A041025 A041026 A041027

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved