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A041024 Numerators of continued fraction convergents to sqrt(17). 5
4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236 (list; graph; refs; listen; history; text; internal format)
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) = (1/2)*sqrt(17)*{[4+sqrt(17)]^n-[4-sqrt(17)]^n}+2*{[4+sqrt(17)]^n+[4-sqrt(17)]^n}, with n>=0. [Paolo P. Lava, Nov 28 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, A041025.

Sequence in context: A213168 A203212 A088317 * A257068 A246806 A202765

Adjacent sequences:  A041021 A041022 A041023 * A041025 A041026 A041027

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 20 16:53 EDT 2017. Contains 289628 sequences.