OFFSET
0,2
COMMENTS
Lim_{n->infinity} a(n)/a(n-1) = 3 + 2*sqrt(2).
REFERENCES
A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for linear recurrences with constant coefficients, signature (6,-1).
FORMULA
a(n) = ((3+2*sqrt(2))^n - (3-2*sqrt(2))^n) * (3/(2*sqrt(2)));
a(n) = 6*a(n-1) - a(n-2).
a(n) = 6*A001109(n).
G.f.: 6x/(1-6x+x^2). - Philippe Deléham, Nov 17 2008
MATHEMATICA
LinearRecurrence[{6, -1}, {0, 6}, 30] (* Harvey P. Dale, Nov 28 2012 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gregory V. Richardson, Oct 15 2002
STATUS
approved