A097766 Pell equation solutions (11*a(n))^2 - 122*b(n)^2 = -1 with b(n):=A097767(n), n >= 0.

1, 487, 236681, 115026479, 55902632113, 27168564180439, 13203866289061241, 6417051847919582687, 3118673994222628124641, 1515669144140349348992839, 736612085378215560982395113, 357991957824668622288095032079, 173983354890703572216453203195281

Offset

0,2

Links

Michael De Vlieger, Table of n, a(n) for n = 0..372

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 (486,-1).

Formula

G.f.: (1 + x)/(1 - 2*243*x + x^2).

a(n) = S(n, 2*243) + S(n-1, 2*243) = S(2*n, 2*sqrt(122)), with Chebyshev polynomials of the second 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, 11*i)/(11*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

a(n) = 486*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=487. - Philippe Deléham, Nov 18 2008

a(n) = (1/11)*sinh((2*n + 1)*arcsinh(11)). - Bruno Berselli, Apr 03 2018

Example

(x,y) = (11*1=11;1), (5357=11*487;485), (2603491=11*236681;235709), ... give the positive integer solutions to x^2 - 122*y^2 =-1.

Mathematica

LinearRecurrence[{486, -1}, {1, 487}, 12] (* Ray Chandler, Aug 12 2015 *)

Prog

(PARI) x='x+O('x^99); Vec((1+x)/(1-2*243*x+x^2)) \\ Altug Alkan, Apr 05 2018

Crossrefs

Cf. A097765 for S(n, 2*243).

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

Sequence in context: A236424 A235839 A203873 * A214806 A126819 A045011

Adjacent sequences: A097763 A097764 A097765 * A097767 A097768 A097769

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved