A075841 Numbers k such that 2*k^2 - 9 is a square.

3, 15, 87, 507, 2955, 17223, 100383, 585075, 3410067, 19875327, 115841895, 675176043, 3935214363, 22936110135, 133681446447, 779152568547, 4541233964835, 26468251220463, 154268273357943, 899141388927195

Offset

1,1

Comments

Lim. n-> Inf. a(n)/a(n-1) = 3 + 2*sqrt(2).

Positive values of x (or y) satisfying x^2 - 6*x*y + y^2 + 36 = 0. - Colin Barker, Feb 08 2014

For each member t of the sequence there exists a nonnegative r such that t^2 = r^2 + (r+3)^2. The r values are in A241976. Example: 87^2 = 60^2 + 63^2. - Bruno Berselli, Jul 10 2017

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

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Tanya Khovanova, Recursive Sequences

J. J. O'Connor and E. F. Robertson, Pell's Equation

Eric Weisstein's World of Mathematics, Pell Equation.

Index entries for linear recurrences with constant coefficients, signature (6,-1).

Formula

a(n) = 3*sqrt(2)/4*((1+sqrt(2))^(2*n-1)-(1-sqrt(2))^(2*n-1)) = 6*a(n-1) - a(n-2).

G.f.: 3*x*(1-x)/(1-6*x+x^2). - Philippe Deléham, Nov 17 2008

a(n) = 3*A001653(n). - R. J. Mathar, Sep 27 2014

Mathematica

CoefficientList[Series[3 (1 - x)/(1 - 6 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 11 2014 *)
LinearRecurrence[{6, -1}, {3, 15}, 20] (* Harvey P. Dale, Jun 05 2023 *)

Prog

(PARI) isok(n) = issquare(2*n^2-9); \\ Michel Marcus, Jul 10 2017

Crossrefs

Sequence in context: A355097 A180677 A220875 * A152596 A278392 A370287

Adjacent sequences: A075838 A075839 A075840 * A075842 A075843 A075844

Keyword

nonn,easy

Author

Gregory V. Richardson, Oct 14 2002

Status

approved