A098258 Chebyshev polynomials S(n,531) + S(n-1,531) with Diophantine property.

1, 532, 282491, 150002189, 79650879868, 42294467207719, 22458282436418921, 11925305679271239332, 6332314857410591666371, 3362447263979344903603669, 1785453164858174733221881868, 948072268092426803995915668239

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(23*a(n))^2 - 533*b(n)^2 = -4 with b(n)=A098259(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

• Muniru A Asiru, Table of n, a(n) for n = 0..300
• Tanya Khovanova, Recursive Sequences
• Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
• Index entries for linear recurrences with constant coefficients, signature (531, -1).
• Index entries for sequences related to Chebyshev polynomials.

Programs

GAP

a:=[1,532];; for n in [3..12] do a[n]:=531*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Apr 29 2019

Mathematica

LinearRecurrence[{531,-1},{1,532},20] (* Harvey P. Dale, Oct 09 2018 *)

Formula

a(n) = S(n, 531) + S(n-1, 531) = S(2*n, sqrt(533)), 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)=A098257(n).

a(n)= (-2/23)*i*((-1)^n)*T(2*n+1, 23*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

G.f.: (1+x)/(1-531*x+x^2).

a(n) = 531*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=532. - Philippe Deléham, Nov 18 2008