A075848 Numbers k such that 2*k^2 + 9 is a square.

0, 6, 36, 210, 1224, 7134, 41580, 242346, 1412496, 8232630, 47983284, 279667074, 1630019160, 9500447886, 55372668156, 322735561050, 1881040698144, 10963508627814, 63900011068740, 372436557784626, 2170719335639016

Offset: 0

Author

Gregory V. Richardson, Oct 15 2002

Keywords

• nonn
• easy

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).

Programs

Mathematica

LinearRecurrence[{6,-1},{0,6},30] (* Harvey P. Dale, Nov 28 2012 *)

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