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

Author

Gregory V. Richardson, Oct 14 2002

Keywords

• nonn
• easy

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).

Programs

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 *)

PARI

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

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