A097768 - OEIS

%I #13 Jun 20 2020 10:14:34

%S 1,578,334083,193099396,111611116805,64511032413894,37287265124113927,

%T 21551974730705435912,12457004107082617843209,

%U 7200126821919022407938890,4161660846065087869170835211,2405432768898798869358334813068

%N Chebyshev U(n,x) polynomial evaluated at x=289=2*12^2+1.

%C Used to form integer solutions of Pell equation a^2 - 145*b^2 =-1. See A097769 with A097770.

%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 (578,-1).

%F a(n) = 2*289*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

%F a(n) = S(n, 2*289)= U(n, 289), Chebyshev's polynomials of the second kind. See A049310.

%F G.f.: 1/(1-2*289*x+x^2).

%F a(n)= sum((-1)^k*binomial(n-k, k)*578^(n-2*k), k=0..floor(n/2)), n>=0.

%F a(n) = ((289+24*sqrt(145))^(n+1) - (289-24*sqrt(145))^(n+1))/(48*sqrt(145)), n>=0.

%t LinearRecurrence[{578, -1},{1, 578},12] (* _Ray Chandler_, Aug 12 2015 *)

%o (PARI) a(n) = polchebyshev(n, 2, 289); \\ _Michel Marcus_, Jun 20 2020

%Y Cf. A049310, A097769, A097770.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 31 2004