A041024 - OEIS

%I #53 Dec 26 2023 07:44:22

%S 4,33,268,2177,17684,143649,1166876,9478657,76996132,625447713,

%T 5080577836,41270070401,335241141044,2723199198753,22120834731068,

%U 179689877047297,1459639851109444,11856808685922849,96314109338492236

%N Numerators of continued fraction convergents to sqrt(17).

%C 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).

%C 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

%H Vincenzo Librandi, <a href="/A041024/b041024.txt">Table of n, a(n) for n = 0..200</a>

%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent sequences in the equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,1).

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

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

%F 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.

%F a(n) = 8*a(n-1) + a(n-2), n > 1. - _Philippe Deléham_, Nov 20 2008

%F a(n) = ((4 + sqrt(17))^n + (4 - sqrt(17))^n)/2. - _Sture Sjöstedt_, Dec 08 2011

%t Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[17],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 17 2011*)

%t LinearRecurrence[{8, 1}, {4, 33}, 25] (* _Sture Sjöstedt_, Dec 07 2011 *)

%t CoefficientList[Series[(4 + x)/(1 - 8 x - x^2), {x, 0, 30}], x]  (* Vincenzo Librandi_, Oct 28 2013 *)

%Y Cf. A010473, A040012, A041025.

%K nonn,cofr,frac,easy

%O 0,1

%A _N. J. A. Sloane_