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A098259 First differences of Chebyshev polynomials S(n,531)=A098257(n) with Diophantine property. 3
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

Table of n, a(n) for n=0..11.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (531, -1).

Index entries for sequences related to Chebyshev polynomials.

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 . [From 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), ...

CROSSREFS

Sequence in context: A031724 A031611 A251004 * A174761 A031967 A031521

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 April 30 12:33 EDT 2017. Contains 285668 sequences.