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A097840
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Chebyshev polynomials S(n,83) + S(n-1,83) with Diophantine property.
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3
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1, 84, 6971, 578509, 48009276, 3984191399, 330639876841, 27439125586404, 2277116783794691, 188973253929372949, 15682502959354160076, 1301458772372465913359, 108005395603955316648721
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OFFSET
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0,2
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COMMENTS
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(9*a(n))^2 - 85*b(n)^2 = -4 with b(n)=A097841(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..12.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n)= S(n, 83) + S(n-1, 83) = S(2*n, sqrt(85)), 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, 83)=A097839(n).
a(n)= (-2/9)*I*((-1)^n)*T(2*n+1, 9*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-83*x+x^2).
a(n)=83*a(n-1)-a(n-2) for n>1 ; a(0)=1, a(1)=84. [From Philippe DELEHAM, Nov 18 2008]
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EXAMPLE
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All positive solutions of Pell equation x^2 - 85*y^2 = -4 are
(9=9*1,1), (756=9*84,82), (62739=9*6971,6805), (5206581=9*578509,564733), ...
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CROSSREFS
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Sequence in context: A075906 A075909 A132052 * A224177 A223869 A224389
Adjacent sequences: A097837 A097838 A097839 * A097841 A097842 A097843
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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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