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%I #37 Sep 08 2022 08:45:14
%S 1,327,106601,34751599,11328914673,3693191431799,1203969077851801,
%T 392490226188255327,127950609768293384801,41711506294237455189799,
%U 13597823101311642098489673,4432848619521301086652443599,1445095052140842842606598123601
%N Pell equation solutions (9*a(n))^2 - 82*b(n)^2 = -1 with b(n):=A097739(n), n >= 0.
%H Indranil Ghosh, <a href="/A097738/b097738.txt">Table of n, a(n) for n = 0..397</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 (326,-1).
%F G.f.: (1 + x)/(1 - 2*163*x + x^2).
%F a(n) = S(n, 2*163) + S(n-1, 2*163) = S(2*n, 2*sqrt(82)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x) = U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
%F a(n) = ((-1)^n)*T(2*n+1, 9*i)/(9*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
%F a(n) = 326*a(n-1) - a(n-2), n>1; a(0)=1, a(1)=327. - _Philippe Deléham_, Nov 18 2008
%F a(n) = (1/9)*sinh((2*n + 1)*arcsinh(9)). - _Bruno Berselli_, Apr 03 2018
%e (x,y) = (9*1=9;1), (2943=9*327;325), (959409=9*106601;105949), ... give the positive integer solutions to x^2 - 82*y^2 =-1.
%t LinearRecurrence[{326, -1}, {1, 327}, 12] (* _Ray Chandler_, Aug 12 2015 *)
%o (PARI) x='x+O('x^99); Vec((1+x)/(1-2*163*x+x^2)) \\ _Altug Alkan_, Apr 05 2018
%o (Magma) a:=[1,327]; [n le 2 select a[n] else 326*Self(n-1) - Self(n-2): n in [1..13]]; // _Marius A. Burtea_, Jan 23 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 13); Coefficients(R!( (1 + x)/(1 - 2*163*x + x^2))); // _Marius A. Burtea_, Jan 23 2020
%Y Cf. A097737 for S(n, 2*163).
%Y Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
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