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

0,2

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

Sequence in context: A269406 A269603 A027910 * A096979 A269464 A123887

Adjacent sequences: A075845 A075846 A075847 * A075849 A075850 A075851

Keyword

nonn,easy

Author

Gregory V. Richardson, Oct 15 2002

Status

approved