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%I #32 Sep 08 2022 08:45:14
%S 1,82,6805,564733,46866034,3889316089,322766369353,26785719340210,
%T 2222891938868077,184473245206710181,15309056460218076946,
%U 1270467212952893676337,105433469618629957059025
%N First differences of Chebyshev polynomials S(n,83) = A097839(n) with Diophantine property.
%C (9*b(n))^2 - 85*a(n)^2 = -4 with b(n)=A097840(n) give all positive solutions of this Pell equation.
%H Indranil Ghosh, <a href="/A097841/b097841.txt">Table of n, a(n) for n = 0..520</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 (83, -1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = ((-1)^n)*S(2*n, 9*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
%F G.f.: (1-x)/(1 - 83*x + x^2).
%F a(n) = S(n, 83) - S(n-1, 83) = T(2*n+1, sqrt(85)/2)/(sqrt(85)/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) = 83*a(n-1) - a(n-2) for n > 1, a(0)=1, a(1)=82. - _Philippe Deléham_, Nov 18 2008
%e All positive solutions of Pell equation x^2 - 85*y^2 = -4 are (9=9*1,1), (756=9*84,82), (62739=9*6971,6805), (5206581=9*578509,564733), ...
%t CoefficientList[Series[(1-x)/(1-83x+x^2), {x, 0, 20}], x] (* _Michael De Vlieger_, Feb 08 2017 *)
%t LinearRecurrence[{83,-1}, {1,82}, 20] (* _G. C. Greubel_, Jan 13 2019 *)
%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-83*x+x^2)) \\ _G. C. Greubel_, Jan 13 2019
%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-83*x+x^2) )); // _G. C. Greubel_, Jan 13 2019
%o (Sage) ((1-x)/(1-83*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 13 2019
%o (GAP) a:=[1,82];; for n in [3..20] do a[n]:=83*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 13 2019
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
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