OFFSET
0,2
COMMENTS
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..548
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 (66, -1).
FORMULA
G.f.: (1 + x)/(1 - 66*x + x^2).
a(n) = 66*a(n-1) - a(n-2) for n>=1, a(-1)=-1, a(0)=1.
a(n) = S(2*n, 2*sqrt(17)) = -i*((-1)^n)*T(2*n+1, 4*i)/4 = S(n, 66) + S(n-1, 66) with i^2=-1 and S(n, x), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
a(n) = A041024(2*n)/4.
a(n) = (1/4)*sinh((2*n + 1)*arcsinh(4)). - Bruno Berselli, Apr 03 2018
EXAMPLE
(x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.
MATHEMATICA
LinearRecurrence[{66, -1}, {1, 67}, 20] (* Bruno Berselli, Apr 03 2018 *)
PROG
(PARI) x='x+O('x^99); Vec((1+x)/(1-66*x+x^2)) \\ Altug Alkan, Apr 05 2018
(GAP) a:=[1, 67];; for n in [3..20] do a[n]:=66*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Apr 05 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 10 2003
STATUS
approved