A097769 Pell equation solutions (12*a(n))^2 - 145*b(n)^2 = -1 with b(n):=A097770(n), n >= 0.

1, 579, 334661, 193433479, 111804216201, 64622643530699, 37351776156527821, 21589261995829549839, 12478556081813323279121, 7212583826026105025782099, 4168860972887006891578774101, 2409594429744863957227505648279

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Examples

			(x,y) = (12*1=12;1), (6948=12*579;577), (4015932=12*334661;333505), ... give the positive integer solutions to x^2 - 145*y^2 = -1.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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 (578,-1).

Crossrefs

Cf. A097768 for S(n, 2*289).

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

Programs

Magma

I:=[1, 579]; [n le 2 select I[n] else 578*Self(n-1)-Self(n-2): n in [1..15]]; // Vincenzo Librandi, May 20 2012

Mathematica

LinearRecurrence[{578, -1}, {1, 579}, 20] (* or *) CoefficientList[Series[(1 + x)/(1 - 578 x + x^2), {x, 0, 20}], x] (* Harvey P. Dale, May 15 2011 *)

PARI

x='x+O('x^99); Vec((1+x)/(1-2*289*x+x^2)) \\ Altug Alkan, Apr 05 2018

Formula

G.f.: (1 + x)/(1 - 2*289*x + x^2).

a(n) = S(n, 2*289) + S(n-1, 2*289) = S(2*n, 2*sqrt(145)), 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, 12*i)/(12*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

a(n) = 578*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=579. - Philippe Deléham, Nov 18 2008

a(n) = (1/12)*sinh((2*n + 1)*arcsinh(12)). - Bruno Berselli, Apr 05 2018