A098252 Chebyshev polynomials S(n,363) + S(n-1,363) with Diophantine property.

1, 364, 132131, 47963189, 17410505476, 6319965524599, 2294130074923961, 832762897231873244, 302290637565095063611, 109730668673232276217549, 39831930437745751171906676, 14458881018233034443125905839

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy
• changed

Comments

(19*a(n))^2 - 365*b(n)^2 = -4 with b(n)=A098253(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 365*y^2 = -4 are (19=19*1,1), (6916=19*364,362), (2510489=19*132131,131405),(911300591=19*47963189,47699653), ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..389
• 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 (363,-1).

Crossrefs

Cf. A049310, A053120, A098251, A098253.

Programs

Mathematica

LinearRecurrence[{363,-1},{1,364},20] (* Harvey P. Dale, Feb 03 2015 *)

Formula

a(n) = S(n, 363) + S(n-1, 363) = S(2*n, sqrt(365)), 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, 363)=A098251(n).

a(n) = (-2/19)*i*((-1)^n)*T(2*n+1, 19*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

G.f.: (1+x)/(1-363*x+x^2).

a(n) = 363*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=364. - Philippe Deléham, Nov 18 2008

E.g.f.: exp(363*x/2)*(19*cosh(19*sqrt(365)*x/2) + sqrt(365)*sinh(19*sqrt(365)*x/2))/19. - Stefano Spezia, Aug 23 2025