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

1, 530, 281429, 149438269, 79351439410, 42135464888441, 22373852504322761, 11880473544330497650, 6308509078186989929389, 3349806440043747322007909, 1778740911154151640996270290, 944508074016414477621697516081

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(23*b(n))^2 - 533*a(n)^2 = -4 with b(n)=A098258(n) give all positive solutions of this Pell equation.

Examples

			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), ...
		

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).

Programs

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

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

Mathematica

LinearRecurrence[{531,-1}, {1,530}, 20] (* G. C. Greubel, Aug 01 2019 *)

PARI

my(x='x+O('x^20)); Vec((1-x)/(1-531*x+x^2)) \\ 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

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