A097772 Pell equation solutions (13*a(n))^2 - 170*b(n)^2 = -1 with b(n):=A097771(n), n >= 0.

1, 679, 460361, 312124079, 211619665201, 143477820882199, 97277750938465721, 65954171658458876639, 44716831106684179895521, 30317945536160215510286599, 20555522356685519431794418601, 13936613839887246014541105524879

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Examples

			(x,y) = (13*1=13;1), (8827=13*679;677), (5984693=13*460361;459005), ... give the positive integer solutions to x^2 - 170*y^2 =-1.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..353
• 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 (678,-1).

Crossrefs

Cf. A097771 for S(n, 2*339).

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

Programs

Mathematica

LinearRecurrence[{678, -1}, {1, 679}, 12] (* Ray Chandler, Aug 12 2015 *)

PARI

x='x+O('x^99); Vec((1+x)/(1-2*339*x+x^2)) \\ Altug Alkan, Apr 05 2018

Formula

G.f.: (1 + x)/(1 - 2*339*x + x^2).

a(n) = S(n, 2*339) + S(n-1, 2*339) = S(2*n, 2*sqrt(170)), 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, 13*i)/(13*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

a(n) = 678*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=679. - Philippe Deléham, Nov 18 2008

a(n) = (1/13)*sinh((2*n + 1)*arcsinh(13)). - Bruno Berselli, Apr 05 2018