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Search: id:A098259
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| A098259 |
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First differences of Chebyshev polynomials S(n,531)=A098257(n) with Diophantine property. |
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4
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| 1, 530, 281429, 149438269, 79351439410, 42135464888441, 22373852504322761, 11880473544330497650, 6308509078186989929389, 3349806440043747322007909, 1778740911154151640996270290, 944508074016414477621697516081
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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(23*b(n))^2 - 533*a(n)^2 = -4 with b(n)=A098258(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)= ((-1)^n)*S(2*n, 23*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-531*x+x^2).
a(n)= S(n, 531) - S(n-1, 531) = T(2*n+1, sqrt(533)/2)/(sqrt(533)/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)=531*a(n-2)-a(n-2), n>1 ; a(0)=1, a(1)=530 . [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 - 533*y^2 = -4 are
(23=23*1,1), (12236=23*532,530), (6497293=23*282491,281429),
(3450050347=23*150002189,149438269), ...
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CROSSREFS
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Sequence in context: A158368 A031724 A031611 this_sequence A031967 A031521 A156772
Adjacent sequences: A098256 A098257 A098258 this_sequence A098260 A098261 A098262
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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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