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Search: id:A097842
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| A097842 |
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Chebyshev polynomials S(n,123) + S(n-1,123) with diophantine property. |
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3
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| 1, 124, 15251, 1875749, 230701876, 28374454999, 3489827263001, 429220378894124, 52790616776714251, 6492816643156958749, 798563656491529211876, 98216836931814936101999
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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(11*a(n))^2 - 5*(5*b(n))^2 = -4 with b(n)=A097843(n) give all positive solutions of this Pell equation.
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LINKS
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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, 123) + S(n-1, 123) = S(2*n, 5*sqrt(5)), 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, 123)=A049670(n+1).
a(n)= (-2/11)*I*((-1)^n)*T(2*n+1, 11*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-123*x+x^2).
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EXAMPLE
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All positive solutions of Pell equation x^2 - 125*y^2 = -4 are
(11=11*1,1), (1364=11*124,122), (167761=11*15251,15005),
(20633239=11*1875749,1845493), ...
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CROSSREFS
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Adjacent sequences: A097839 A097840 A097841 this_sequence A097843 A097844 A097845
Sequence in context: A038859 A024017 A035816 this_sequence A009805 A005080 A013308
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 10 2004
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