A075869 Numbers k such that 5*k^2 - 9 is a square.

3, 51, 915, 16419, 294627, 5286867, 94868979, 1702354755, 30547516611, 548152944243, 9836205479763, 176503545691491, 3167227616967075, 56833593559715859, 1019837456457918387, 18300240622682815107

Offset: 1

Author

Gregory V. Richardson, Oct 16 2002

Keywords

• nonn
• easy

Comments

Lim. n-> Inf. a(n)/a(n-1) = phi^6 = 9 + 4*sqrt(5).

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

• 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 (18,-1).

Crossrefs

Cf. 3*A007805.

Programs

Mathematica

LinearRecurrence[{18, -1}, {3, 51}, 20] (* Harvey P. Dale, Dec 27 2018 *)

Formula

a(n) = 3*sqrt(5)/10*((2+sqrt(5))^(2*n-1)-(2-sqrt(5))^(2*n-1)) = 18*a(n-1) - a(n-2).

G.f.: 3*x*(1-x)/(1-18*x+x^2). [Philippe Deléham, Nov 17 2008; corrected by Georg Fischer, May 15 2019]