A075873 40*n^2 + 9 is a square.

0, 1, 2, 9, 40, 77, 342, 1519, 2924, 12987, 57682, 111035, 493164, 2190397, 4216406, 18727245, 83177404, 160112393, 711142146, 3158550955, 6080054528, 27004674303, 119941758886, 230881959671, 1025466481368, 4554628286713

Offset

1,3

Comments

Lim. n-> Inf. a(n)/a(n-3) = 19 + 6*Sqrt(10). Lim. n-> Inf. a(3*k)/a(3*k-1) = (11 + 2*Sqrt(10))/9. Lim. n-> Inf. a(3*k+1)/a(3*k) = (7 + 2*Sqrt(10))/3. Lim. n-> Inf. a(3*k+2)/a(3*k+1) = (7 + 2*Sqrt(10))/3.

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

Table of n, a(n) for n=1..26.

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 (0,0,38,0,0,-1)

Formula

G.f.: x^2*(x^5+2x^4+9x^3+2x^2+x)/(x^6-38x^3+1).

a(n) = A075836(n)/2.

Mathematica

LinearRecurrence[{0, 0, 38, 0, 0, -1}, {0, 1, 2, 9, 40, 77}, 30] (* Harvey P. Dale, Sep 05 2020 *)

Crossrefs

Sequence in context: A020002 A346687 A120700 * A124722 A317129 A220309

Adjacent sequences: A075870 A075871 A075872 * A075874 A075875 A075876

Keyword

nonn

Author

Gregory V. Richardson, Oct 16 2002

Status

approved