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A098250
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First differences of Chebyshev polynomials S(n,291)=A098248(n) with Diophantine property.
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4
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1, 290, 84389, 24556909, 7145976130, 2079454496921, 605114112627881, 176086127320216450, 51240457936070359069, 14910797173269154272629, 4338990736963387822975970, 1262631393659172587331734641
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OFFSET
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0,2
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COMMENTS
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(17*b(n))^2 - 293*a(n)^2 = -4 with b(n)=A098249(n) give all positive solutions of this Pell equation.
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LINKS
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Table of n, a(n) for n=0..11.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n)= ((-1)^n)*S(2*n, 17*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-291*x+x^2).
a(n)= S(n, 291) - S(n-1, 291) = T(2*n+1, sqrt(293)/2)/(sqrt(293)/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)=291*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=290 . [From Philippe DELEHAM, Nov 18 2008]
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EXAMPLE
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All positive solutions of Pell equation x^2 - 293*y^2 = -4 are
(17=17*1,1), (4964=17*292,290), (1444507=17*84971,84389),
(420346573=17*24726269,24556909), ...
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CROSSREFS
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Sequence in context: A031605 A186548 A091740 * A031515 A090890 A123913
Adjacent sequences: A098247 A098248 A098249 * A098251 A098252 A098253
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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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