OFFSET
0,2
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 linear recurrences with constant coefficients, signature (578,-1).
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
EXAMPLE
(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.
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 *)
PROG
(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
(PARI) x='x+O('x^99); Vec((1+x)/(1-2*289*x+x^2)) \\ Altug Alkan, Apr 05 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 31 2004
STATUS
approved