A097735 Pell equation solutions (8*a(n))^2 - 65*b(n)^2 = -1 with b(n):=A097736(n), n >= 0.

1, 259, 66821, 17239559, 4447739401, 1147499525899, 296050429942541, 76379863425649679, 19705708713387674641, 5083996468190594407699, 1311651383084459969511701, 338400972839322481539611159, 87306139341162115777250167321, 22524645549046986548049003557659

Offset

0,2

Links

Indranil Ghosh, Table of n, a(n) for n = 0..413

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 (258,-1).

Formula

G.f.: (1 + x)/(1 - 2*129*x + x^2).

a(n) = S(n, 2*129) + S(n-1, 2*129) = S(2*n, 2*sqrt(65)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x) = U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).

a(n) = ((-1)^n)*T(2*n+1, 8*i)/(8*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

a(n) = 258*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=259. - Philippe Deléham, Nov 18 2008

a(n) = (1/8)*sinh((2*n + 1)*arcsinh(8)). - Bruno Berselli, Apr 03 2018

Example

(x,y) = (8,1), (2072,257), (534568,66305), ... give the positive integer solutions to x^2 - 65*y^2 =-1.

Mathematica

LinearRecurrence[{258, -1}, {1, 259}, 20] (* Harvey P. Dale, Oct 30 2011 *)

Prog

(PARI) x='x+O('x^99); Vec((1+x)/(1-2*129*x+x^2)) \\ Altug Alkan, Apr 05 2018

Crossrefs

Cf. A097731 for S(n, 2*129).

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Sequence in context: A229433 A022221 A121918 * A063485 A252248 A214471

Adjacent sequences: A097732 A097733 A097734 * A097736 A097737 A097738

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved