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A098261
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Chebyshev polynomials S(n,627) + S(n-1,627) with Diophantine property.
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2
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1, 628, 393755, 246883757, 154795721884, 97056670737511, 60854377756697513, 38155597796778603140, 23923498964202427471267, 14999995694957125245881269, 9404973377239153326740084396
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OFFSET
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0,2
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COMMENTS
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(25*a(n))^2 - 629*b(n)^2 = -4 with b(n)=A098262(n) give all positive solutions of this Pell equation.
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LINKS
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Table of n, a(n) for n=0..10.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n)= S(n, 627) + S(n-1, 627) = S(2*n, sqrt(629)), 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, 531)=A098260(n).
a(n)= (-2/25)*I*((-1)^n)*T(2*n+1, 25*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-627*x+x^2).
a(n)=627*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=628 . [From Philippe DELEHAM, Nov 18 2008]
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EXAMPLE
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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
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Sequence in context: A188362 A098260 A224603 * A177421 A173423 A110904
Adjacent sequences: A098258 A098259 A098260 * A098262 A098263 A098264
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Sep 10 2004
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STATUS
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approved
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