A098255 Chebyshev polynomials S(n,443) + S(n-1,443) with Diophantine property.

1, 444, 196691, 87133669, 38600018676, 17099721139799, 7575137864912281, 3355768974435000684, 1486598080536840390731, 658559593908845858093149, 291740413503538178294874276, 129240344622473504138771211119

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(21*a(n))^2 - 445*b(n)^2 = -4 with b(n)=A098256(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 445*y^2 = -4 are (21=21*1,1), (9324=21*444,442), (4130511=21*196691,195805),(1829807049=21*87133669,86741173), ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..377
• 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 (443, -1).
• Index entries for sequences related to Chebyshev polynomials.

Programs

Mathematica

LinearRecurrence[{443,-1},{1,444},12] (* Indranil Ghosh, Feb 18 2017 *)

Formula

a(n) = S(n, 443) + S(n-1, 443) = S(2*n, sqrt(445)), 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, 443)=A098254(n).

a(n) = (-2/21)*i*((-1)^n)*T(2*n+1, 21*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

G.f.: (1+x)/(1-443*x+x^2).

a(n) = 443*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=444. - Philippe Deléham, Nov 18 2008