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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

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).

Crossrefs

Cf. A010473, A040012, A041025.

Programs

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 *)

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