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%I #27 Sep 08 2022 08:45:14
%S 1,442,195805,86741173,38426143834,17022694977289,7541015448795193,
%T 3340652821121293210,1479901658741284096837,655593094169567733605581,
%U 290426260815459764703175546,128658177948154506195773161297,56995282404771630784962807279025,25248781447135884283232327851446778
%N First differences of Chebyshev polynomials S(n,443)=A098254(n) with Diophantine property.
%C (21*b(n))^2 - 445*a(n)^2 = -4 with b(n)=A098255(n) give all positive solutions of this Pell equation.
%H Indranil Ghosh, <a href="/A098256/b098256.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/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 (443, -1).
%F G.f.: (1 - x)/(1 - 443*x + x^2).
%F a(n) = ((-1)^n)*S(2*n, 21*i) with the imaginary unit i and the S(n, x)=U(n, x/2) Chebyshev polynomials.
%F a(n) = S(n, 443) - S(n-1, 443) = T(2*n+1, sqrt(445)/2)/(sqrt(445)/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) = 443*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=442. - _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}, {1,442}, 20] (* _G. C. Greubel_, Aug 01 2019 *)
%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-443*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019
%o (Magma) I:=[1,442]; [n le 2 select I[n] else 443*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) ((1-x)/(1-443*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o (GAP) a:=[1,442];; for n in [3..20] do a[n]:=443*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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