A098255 - OEIS

%I #29 Jan 23 2020 00:58:00

%S 1,444,196691,87133669,38600018676,17099721139799,7575137864912281,

%T 3355768974435000684,1486598080536840390731,658559593908845858093149,

%U 291740413503538178294874276,129240344622473504138771211119

%N Chebyshev polynomials S(n,443) + S(n-1,443) with Diophantine property.

%C (21*a(n))^2 - 445*b(n)^2 = -4 with b(n)=A098256(n) give all positive solutions of this Pell equation.

%H Indranil Ghosh, <a href="/A098255/b098255.txt">Table of n, a(n) for n = 0..377</a>

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

%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (443, -1).

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

%F a(n) = S(n, 443) + S(n-1, 443) = S(2*n, sqrt(445)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 443)=A098254(n).

%F a(n) = (-2/21)*i*((-1)^n)*T(2*n+1, 21*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

%F G.f.: (1+x)/(1-443*x+x^2).

%F a(n) = 443*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=444. - _Philippe Deléham_, Nov 18 2008

%e All positive solutions of Pell equation x^2 - 445*y^2 = -4 are (21=21*1,1), (9324=21*444,442), (4130511=21*196691,195805),(1829807049=21*87133669,86741173), ...

%t LinearRecurrence[{443,-1},{1,444},12] (* _Indranil Ghosh_, Feb 18 2017 *)

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Sep 10 2004