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A097739
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Pell equation solutions (9*b(n))^2 - 82*a(n)^2 = -1 with b(n):=A097738(n), n>=0.
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5
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1, 325, 105949, 34539049, 11259624025, 3670602893101, 1196605283526901, 390089651826876625, 127168029890278252849, 41456387654578883552149, 13514655207362825759747725, 4405736141212626618794206201, 1436256467380108914901151473801, 468215202629774293631156586252925
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OFFSET
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0,2
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LINKS
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Indranil Ghosh, Table of n, a(n) for n = 0..397
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (326, -1).
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FORMULA
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a(n) = S(n, 2*163) - S(n-1, 2*163) = T(2*n+1, sqrt(82))/sqrt(82), 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.
a(n) = ((-1)^n)*S(2*n, 18*I) with the imaginary unit I and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1- 326*x+x^2).
a(n) = 326*a(n-1) - a(n-2), n>1; a(0)=1, a(1)=325. - Philippe Deléham, Nov 18 2008
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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, 325}, 12] (* Ray Chandler, Aug 12 2015 *)
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PROG
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(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-326*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(MAGMA) I:=[1, 325]; [n le 2 select I[n] else 326*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-326*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 325];; for n in [3..20] do a[n]:=326*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
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CROSSREFS
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Cf. A097737 for S(n, 326).
Row 9 of array A188647.
Sequence in context: A121000 A290949 A048909 * A203188 A048918 A274307
Adjacent sequences: A097736 A097737 A097738 * A097740 A097741 A097742
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Aug 31 2004
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STATUS
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approved
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