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 A098259 First differences of Chebyshev polynomials S(n,531)=A098257(n) with Diophantine property. 4
 1, 530, 281429, 149438269, 79351439410, 42135464888441, 22373852504322761, 11880473544330497650, 6308509078186989929389, 3349806440043747322007909, 1778740911154151640996270290, 944508074016414477621697516081 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS (23*b(n))^2 - 533*a(n)^2 = -4 with b(n)=A098258(n) give all positive solutions of this Pell equation. LINKS G. C. Greubel, Table of n, a(n) for n = 0..365 Tanya Khovanova, Recursive Sequences Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16. Index entries for sequences related to Chebyshev polynomials. Index entries for linear recurrences with constant coefficients, signature (531,-1). FORMULA 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. - Philippe Deléham, Nov 18 2008 EXAMPLE 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), ... MATHEMATICA LinearRecurrence[{531, -1}, {1, 530}, 20] (* G. C. Greubel, Aug 01 2019 *) PROG (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-531*x+x^2)) \\ G. C. Greubel, Aug 01 2019 (Magma) I:=[1, 530]; [n le 2 select I[n] else 531*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019 (Sage) ((1-x)/(1-531*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019 (GAP) a:=[1, 530];; for n in [3..20] do a[n]:=531*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019 CROSSREFS Sequence in context: A250795 A031724 A251004 * A174761 A031967 A345387 Adjacent sequences: A098256 A098257 A098258 * A098260 A098261 A098262 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Sep 10 2004 STATUS approved

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Last modified August 13 22:54 EDT 2024. Contains 375146 sequences. (Running on oeis4.)