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

Author

Gregory V. Richardson, Oct 16 2002

Keywords

• nonn
• easy

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

• 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)

Programs

Mathematica

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

PARI

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,0,0,38,0,0]^(n-1)*[0;1;2;9;40;77])[1,1] \\ Charles R Greathouse IV, Jul 09 2024

Formula

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

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