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A098262
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First differences of Chebyshev polynomials S(n,627)=A098260(n) with Diophantine property.
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3
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1, 626, 392501, 246097501, 154302740626, 96747572275001, 60660573513685001, 38034082845508220626, 23847309283560140647501, 14952224886709362677762501, 9375021156657486838816440626
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| (25*b(n))^2 - 629*a(n)^2 = -4 with b(n)=A098261(n) give all positive solutions of this Pell equation.
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n)= ((-1)^n)*S(2*n, 25*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-627*x+x^2).
a(n)= S(n, 627) - S(n-1, 627) = T(2*n+1, sqrt(629)/2)/(sqrt(629)/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)=627*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=626 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
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EXAMPLE
| All positive solutions of Pell equation x^2 - 629*y^2 = -4 are
(25=25*1,1), (15700=25*628,626), (9843875=25*393755,392501),
(6172093925=25*246883757,246097501), ...
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CROSSREFS
| Sequence in context: A158383 A031613 A031728 * A031523 A129974 A031703
Adjacent sequences: A098259 A098260 A098261 * A098263 A098264 A098265
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 10 2004
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