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A097738
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Pell equation solutions (9*a(n))^2 - 82*b(n)^2 = -1 with b(n):=A097739(n), n >= 0.
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4
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1, 327, 106601, 34751599, 11328914673, 3693191431799, 1203969077851801, 392490226188255327, 127950609768293384801, 41711506294237455189799, 13597823101311642098489673, 4432848619521301086652443599, 1445095052140842842606598123601
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: (1 + x)/(1 - 2*163*x + x^2).
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).
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.
a(n) = 326*a(n-1) - a(n-2), n>1; a(0)=1, a(1)=327. - Philippe Deléham, Nov 18 2008
a(n) = (1/9)*sinh((2*n + 1)*arcsinh(9)). - Bruno Berselli, Apr 03 2018
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EXAMPLE
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(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.
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MATHEMATICA
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LinearRecurrence[{326, -1}, {1, 327}, 12] (* Ray Chandler, Aug 12 2015 *)
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PROG
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(PARI) x='x+O('x^99); Vec((1+x)/(1-2*163*x+x^2)) \\ Altug Alkan, Apr 05 2018
(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
(Magma) R<x>:=PowerSeriesRing(Integers(), 13); Coefficients(R!( (1 + x)/(1 - 2*163*x + x^2))); // Marius A. Burtea, Jan 23 2020
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CROSSREFS
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Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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