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Search: id:A098249
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| A098249 |
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Chebyshev polynomials S(n,291) + S(n-1,291) with Diophantine property. |
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+0
3
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| 1, 292, 84971, 24726269, 7195259308, 2093795732359, 609287362857161, 177300528795701492, 51593844592186277011, 15013631475797410908709, 4368915165612454388157308, 1271339299561748429542867919
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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(17*a(n))^2 - 293*b(n)^2 = -4 with b(n)=A098250(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)= (-2/17)*I*((-1)^n)*T(2*n+1, 17*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-291*x+x^2).
a(n)= S(n, 291) + S(n-1, 291) = S(2*n, sqrt(293)), 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)=291*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=292 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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: A098248 A048956 A043439 this_sequence A155140 A142284 A142784
Adjacent sequences: A098246 A098247 A098248 this_sequence A098250 A098251 A098252
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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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