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 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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 31 2004
STATUS
approved