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A097835
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First differences of Chebyshev polynomials S(n,27) = A097781(n) with Diophantine property.
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5
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1, 26, 701, 18901, 509626, 13741001, 370497401, 9989688826, 269351100901, 7262490035501, 195817879857626, 5279820266120401, 142359329305393201, 3838422070979496026, 103495036587140999501, 2790527565781827490501
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OFFSET
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0,2
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COMMENTS
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(5*b(n))^2 - 29*a(n)^2 = -4 with b(n)=A097834(n) give all positive solutions of this Pell equation.
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LINKS
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FORMULA
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a(n) = ((-1)^n)*S(2*n, 5*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-27*x+x^2).
a(n) = S(n, 27) - S(n-1, 27) = T(2*n+1, sqrt(29)/2)/(sqrt(29)/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.
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EXAMPLE
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All positive solutions of Pell equation x^2 - 29*y^2 = -4 are (5=5*1,1), (140=5*28,26), (3775=5*755,701), (101785=5*20357,18901), ...
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MATHEMATICA
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LinearRecurrence[{27, -1}, {1, 26}, 30] (* Harvey P. Dale, May 31 2013 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec((1-x)/(1-27*x+x^2)) \\ G. C. Greubel, Jan 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-27*x+x^2) )); // G. C. Greubel, Jan 12 2019
(Sage) ((1-x)/(1-27*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019
(GAP) a:=[1, 26];; for n in [3..30] do a[n]:=27*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 12 2019
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CROSSREFS
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Cf. similar sequences listed in A238379.
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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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