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%I #31 Sep 08 2022 08:45:14
%S 1,290,84389,24556909,7145976130,2079454496921,605114112627881,
%T 176086127320216450,51240457936070359069,14910797173269154272629,
%U 4338990736963387822975970,1262631393659172587331734641,367421396564082259525711804561,106918363768754278349394803392610
%N First differences of Chebyshev polynomials S(n,291)=A098248(n) with Diophantine property.
%C (17*b(n))^2 - 293*a(n)^2 = -4 with b(n)=A098249(n) give all positive solutions of this Pell equation.
%H Indranil Ghosh, <a href="/A098250/b098250.txt">Table of n, a(n) for n = 0..405</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 (291,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = ((-1)^n)*S(2*n, 17*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
%F G.f.: (1-x)/(1-291*x+x^2).
%F a(n) = S(n, 291) - S(n-1, 291) = T(2*n+1, sqrt(293)/2)/(sqrt(293)/2), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.
%F a(n) = 291*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=290. - _Philippe Deléham_, Nov 18 2008
%e All positive solutions of Pell equation x^2 - 293*y^2 = -4 are (17=17*1,1), (4964=17*292,290), (1444507=17*84971,84389), (420346573=17*24726269,24556909), ...
%t LinearRecurrence[{291,-1}, {1,290}, 20] (* _G. C. Greubel_, Aug 01 2019 *)
%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-291*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019
%o (Magma) I:=[1,290]; [n le 2 select I[n] else 291*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) ((1-x)/(1-291*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o (GAP) a:=[1,290];; for n in [3..20] do a[n]:=291*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
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