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A098246
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Chebyshev polynomials S(n,227) + S(n-1,227) with Diophantine property.
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4
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1, 228, 51755, 11748157, 2666779884, 605347285511, 137411167031113, 31191729568777140, 7080385200945379667, 1607216248885032407269, 364831008111701411070396, 82815031625107335280572623
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OFFSET
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0,2
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COMMENTS
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(15*a(n))^2 - 229*b(n)^2 = -4 with b(n)=A098247(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, 227) + S(n-1, 227) = S(2*n, sqrt(229)), 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, 227)=A098245(n).
a(n) = (-2/15)*i*((-1)^n)*T(2*n+1, 15*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-227*x+x^2).
a(n) = 227*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=228. [Philippe Deléham, Nov 18 2008]
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EXAMPLE
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All positive solutions of Pell equation x^2 - 229*y^2 = -4 are (15=15*1,1), (3420=15*228,226), (776325=15*51755,51301), (176222355=15*11748157,11645101), ...
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MATHEMATICA
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LinearRecurrence[{227, -1}, {1, 228}, 20] (* Harvey P. Dale, May 29 2014 *)
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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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