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Search: id:A097845
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| A097845 |
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Chebyshev polynomials S(n,171) + S(n-1,171) with diophantine property. |
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+0
3
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| 1, 172, 29411, 5029109, 859948228, 147046117879, 25144026209081, 4299481435634972, 735186181467371131, 125712537549484828429, 21496108734780438290228, 3675708881109905462800559
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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(13*a(n))^2 - 173*b(n)^2 = -4 with b(n)=A098244(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 relate d to Chebyshev polynomials.
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FORMULA
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a(n)= S(n, 171) + S(n-1, 171) = S(2*n, sqrt(173)), 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, 171)=A097844(n).
a(n)= (-2/13)*I*((-1)^n)*T(2*n+1, 13*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-171*x+x^2).
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EXAMPLE
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All positive solutions of Pell equation x^2 - 173*y^2 = -4 are
(13=13*1,1), (2236=13*172,170), (382343=13*29411,29069),
(65378417=13*5029109,4970629), ...
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CROSSREFS
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Sequence in context: A056132 A043387 A035828 this_sequence A140002 A119567 A142436
Adjacent sequences: A097842 A097843 A097844 this_sequence A097846 A097847 A097848
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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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