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A098244
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First differences of Chebyshev polynomials S(n,171)=A097844(n) with Diophantine property.
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5
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1, 170, 29069, 4970629, 849948490, 145336221161, 24851643870041, 4249485765555850, 726637214266180309, 124250714153751276989, 21246145483077202184810, 3632966626892047822325521, 621216047053057100415479281, 106224311079445872123224631530
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OFFSET
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0,2
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COMMENTS
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(13*b(n))^2 - 173*a(n)^2 = -4 with b(n)=A097845(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, 13*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-171*x+x^2).
a(n) = S(n, 171) - S(n-1, 171) = T(2*n+1, sqrt(173)/2)/(sqrt(173)/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.
a(n) = 171*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=170. - Philippe Deléham, Nov 18 2008
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EXAMPLE
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All positive solutions of Pell equation x^2 - 173*y^2 = -4 are (13=13*1,1), (2236=13*172,170), (382343=13*29411,29069), (65378417=13*5029109,4970629), ...
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MATHEMATICA
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LinearRecurrence[{171, -1}, {1, 170}, 20] (* G. C. Greubel, Aug 01 2019 *)
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PROG
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(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-171*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 170]; [n le 2 select I[n] else 171*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-171*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 170];; for n in [3..20] do a[n]:=171*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
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CROSSREFS
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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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