OFFSET
1,2
COMMENTS
After initial term this sequence bisects A041025. See A099370 for corresponding x values. a(n+1)/a(n) apparently converges to (4+sqrt(17))^2.
The first solution to the equation x^2 - 17*y^2 = 1 is (X(1); Y(1)) = (1, 0) and the other solutions are defined by: (X(n), Y(n))= (33*X(n-1) + 136*Y(n-1), 8*X(n-1) + 33*Y(n-1)) with n >= 2. - Mohamed Bouhamida, Jan 16 2020
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
Tanya Khovanova, Recursive Sequences.
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for linear recurrences with constant coefficients, signature (66,-1).
FORMULA
a(n) = ((33+8*sqrt(17))^(n-1) - (33-8*sqrt(17))^(n-1))/(2*sqrt(17)).
From Mohamed Bouhamida, Feb 07 2007: (Start)
a(n) = 65*(a(n-1) + a(n-2)) - a(n-3).
a(n) = 67*(a(n-1) - a(n-2)) + a(n-3). (End)
From Philippe Deléham, Nov 18 2008: (Start)
a(n) = 66*a(n-1) - a(n-2) for n > 1; a(1)=0, a(2)=8.
G.f.: 8*x^2/(1 - 66*x + x^2). (End)
E.g.f.: (1/17)*exp(33*x)*(33*sqrt(17)*sinh(8*sqrt(17)*x) + 136*(1 - cosh(8*sqrt(17)*x))). - Stefano Spezia, Feb 08 2020
EXAMPLE
A099370(1)^2 - 17*a(1)^2 = 33^2 - 17*8^2 = 1089 - 1088 = 1.
MATHEMATICA
LinearRecurrence[{66, -1}, {0, 8}, 30] (* Vincenzo Librandi, Dec 18 2011 *)
PROG
(PARI) \\ Program uses fact that continued fraction for sqrt(17) = [4, 8, 8, ...].
print1("0, "); forstep(n=2, 40, 2, v=vector(n, i, if(i>1, 8, 4)); print1(contfracpnqn(v)[2, 1], ", "))
(Magma) I:=[0, 8]; [n le 2 select I[n] else 66*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 18 2011
(Maxima) makelist(expand(((33+8*sqrt(17))^n - (33-8*sqrt(17))^n) /(4*sqrt(17)/2)), n, 0, 16); /* Vincenzo Librandi, Dec 18 2011 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rick L. Shepherd, Jul 31 2006
EXTENSIONS
Offset changed from 0 to 1 and g.f. adapted by Vincenzo Librandi, Dec 18 2011
STATUS
approved