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A097836
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Chebyshev polynomials S(n,51).
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5
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1, 51, 2600, 132549, 6757399, 344494800, 17562477401, 895341852651, 45644872007800, 2326993130545149, 118631004785794799, 6047854250944989600, 308321935793408674801, 15718370871212897425251, 801328592496064360013000
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OFFSET
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0,2
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COMMENTS
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Used for all positive integer solutions of Pell equation x^2 - 53*y^2 = -4. See A097837 with A097838.
a(n-1), with a(-1) := 0, and b(n) := A099368(n) give the proper and improper nonnegative solutions of the Pell equation b(n)^2 - 53*(7*a(n-1))^2 = +4, n >= 0. - Wolfdieter Lang, Jun 27 2013
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LINKS
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Indranil Ghosh, Table of n, a(n) for n = 0..584
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (51,-1).
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n) = S(n, 51)=U(n, 51/2)= S(2*n+1, sqrt(53))/sqrt(53) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n) = 51*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=51.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (51+7*sqrt(53))/2 and am := (51-7*sqrt(53))/2 = 1/ap.
G.f.: 1/(1-51*x+x^2).
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MATHEMATICA
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LinearRecurrence[{51, -1}, {1, 51}, 30] (* G. C. Greubel, Jan 12 2019 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec(1/(1-51*x+x^2)) \\ G. C. Greubel, Jan 12 2019
(MAGMA) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-51*x+x^2) )); // G. C. Greubel, Jan 12 2019
(Sage) (1/(1-51*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019
(GAP) a:=[1, 51];; for n in [2..30] do a[n]:=51*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 12 2019
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CROSSREFS
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Sequence in context: A210177 A210080 A218752 * A267786 A267733 A238603
Adjacent sequences: A097833 A097834 A097835 * A097837 A097838 A097839
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Sep 10 2004
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STATUS
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approved
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