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%I #28 Sep 08 2022 08:45:14
%S 1,785,617009,484968289,381184458145,299610499133681,
%T 235493471134615121,185097568701308351425,145486453505757229604929,
%U 114352167357956481161122769,89880658056900288435412891505
%N Pell equation solutions (14*b(n))^2 - 197*a(n)^2 = -1 with b(n)=A097775(n), n >= 0.
%H Michael De Vlieger, <a href="/A097776/b097776.txt">Table of n, a(n) for n = 0..345</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 (786,-1).
%F a(n) = S(n, 2*393) - S(n-1, 2*393) = T(2*n+1, sqrt(197))/sqrt(197), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
%F a(n) = ((-1)^n)*S(2*n, 28*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
%F G.f.: (1-x)/(1-786*x+x^2).
%F a(n) = 786*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=785. - _Philippe Deléham_, Nov 18 2008
%e (x,y) = (14*1=14;1), (11018=14*787;785), (8660134=14*618581;617009), ... give the positive integer solutions to x^2 - 197*y^2 = -1.
%t CoefficientList[Series[(1-x)/(1-786*x+x^2), {x,0,20}], x] (* _Michael De Vlieger_, Apr 15 2019 *)
%t LinearRecurrence[{786,-1}, {1,785}, 20] (* _G. C. Greubel_, Aug 01 2019 *)
%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-786*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019
%o (Magma) I:=[1,785]; [n le 2 select I[n] else 786*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) ((1-x)/(1-786*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o (GAP) a:=[1,785];; for n in [3..20] do a[n]:=786*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019
%Y Cf. A097774 for S(n, 786).
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
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