A098249 Chebyshev polynomials S(n,291) + S(n-1,291) with Diophantine property.

1, 292, 84971, 24726269, 7195259308, 2093795732359, 609287362857161, 177300528795701492, 51593844592186277011, 15013631475797410908709, 4368915165612454388157308, 1271339299561748429542867919

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(17*a(n))^2 - 293*b(n)^2 = -4 with b(n)=A098250(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 293*y^2 = -4 are (17=17*1,1), (4964=17*292,290), (1444507=17*84971,84389), (420346573=17*24726269,24556909), ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..405
• 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 (291,-1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A049310, A053120, A098250.

Programs

Mathematica

LinearRecurrence[{291,-1},{1,292},20] (* Harvey P. Dale, Jan 01 2020 *)

Formula

a(n) = (-2/17)*i*((-1)^n)*T(2*n+1, 17*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

G.f.: (1+x)/(1-291*x+x^2).

a(n) = S(n, 291) + S(n-1, 291) = S(2*n, sqrt(293)), 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, 227)=A098245(n).

a(n) = 291*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=292. [Philippe Deléham, Nov 18 2008]