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Search: id:A098256
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| A098256 |
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First differences of Chebyshev polynomials S(n,443)=A098254(n) with Diophantine property. |
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4
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| 1, 442, 195805, 86741173, 38426143834, 17022694977289, 7541015448795193, 3340652821121293210, 1479901658741284096837, 655593094169567733605581, 290426260815459764703175546, 128658177948154506195773161297
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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(21*b(n))^2 - 445*a(n)^2 = -4 with b(n)=A098255(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, 21*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-443*x+x^2).
a(n)= S(n, 443) - S(n-1, 443) = T(2*n+1, sqrt(445)/2)/(sqrt(445)/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)=443*a(n-1)-a(n-2),n>1 ; a(0)=1, a(1)=442 . [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 - 445*y^2 = -4 are
(21=21*1,1), (9324=21*444,442), (4130511=21*196691,195805),
(1829807049=21*87133669,86741173), ...
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CROSSREFS
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Sequence in context: A094410 A105922 A018237 this_sequence A105980 A031519 A031699
Adjacent sequences: A098253 A098254 A098255 this_sequence A098257 A098258 A098259
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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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