A098245 - OEIS

%I #18 Jan 15 2019 19:14:58

%S 1,227,51528,11696629,2655083255,602692202256,136808474828857,

%T 31054921093948283,7049330279851431384,1600166918605180975885,

%U 363230841193096230094511,82451800783914239050478112

%N Chebyshev polynomials S(n,227).

%C Used for all positive integer solutions of Pell equation x^2 - 229*y^2 = -4. See A098246 with A098247.

%H Indranil Ghosh, <a href="/A098245/b098245.txt">Table of n, a(n) for n = 0..423</a>

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

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (227, -1).

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

%F a(n) = S(n, 227) = U(n, 227/2) = S(2*n+1, sqrt(229))/sqrt(229) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x).

%F a(n) = 227*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=227; a(-1):=0.

%F a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (227+15*sqrt(229))/2 and am := (227-15*sqrt(229))/2 = 1/ap.

%F G.f.: 1/(1-227*x+x^2).

%t CoefficientList[Series[1/(1 - 227*x + x^2), {x, 0, 15}], x] (* _Wesley Ivan Hurt_, Feb 09 2017 *)

%t LinearRecurrence[{227,-1},{1,227},20] (* _Harvey P. Dale_, Jan 15 2019 *)

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Sep 10 2004