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A098255
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Chebyshev polynomials S(n,443) + S(n-1,443) with Diophantine property.
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3
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1, 444, 196691, 87133669, 38600018676, 17099721139799, 7575137864912281, 3355768974435000684, 1486598080536840390731, 658559593908845858093149, 291740413503538178294874276, 129240344622473504138771211119
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OFFSET
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0,2
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COMMENTS
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(21*a(n))^2 - 445*b(n)^2 = -4 with b(n)=A098256(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) = S(n, 443) + S(n-1, 443) = S(2*n, sqrt(445)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 443)=A098254(n).
a(n) = (-2/21)*i*((-1)^n)*T(2*n+1, 21*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-443*x+x^2).
a(n) = 443*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=444. - Philippe Deléham, Nov 18 2008
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EXAMPLE
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All positive solutions of Pell equation x^2 - 445*y^2 = -4 are (21=21*1,1), (9324=21*444,442), (4130511=21*196691,195805),(1829807049=21*87133669,86741173), ...
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MATHEMATICA
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LinearRecurrence[{443, -1}, {1, 444}, 12] (* Indranil Ghosh, Feb 18 2017 *)
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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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