A098291 Chebyshev polynomials S(n,731) + S(n-1,731) with Diophantine property.

1, 732, 535091, 391150789, 285930691668, 209014944458519, 152789638468485721, 111689016705518603532, 81644518422095630696171, 59682031277535200520297469, 43627483219359809484706753668

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(27*a(n))^2 - 733*b(n)^2 = -4 with b(n)=A098292(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 733*y^2 = -4 are (27=27*1,1), (19764=27*732,730), (14447457=27*535091,533629), (10561071303=27*391150789,390082069), ...
		

Links

• 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 (731,-1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A049310, A053120, A098263, A098292.

Formula

a(n) = S(n, 731) + S(n-1, 731) = S(2*n, sqrt(733)), 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, 731) =A098263(n).

a(n) = (-2/27)*i*((-1)^n)*T(2*n+1, 27*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

G.f.: (1+x)/(1-731*x+x^2).