A228203 x-values in the solution to x^2 - 13y^2 = 27.

12, 40, 220, 768, 14808, 51700, 285520, 996852, 19220772, 67106560, 370604740, 1293913128, 24948547248, 87104263180, 481044667000, 1679498243292, 32383195107132, 113061266501080, 624395607161260, 2179987425879888, 42033362300510088, 146753436814138660

Offset: 1

Author

Colin Barker, Aug 16 2013

Keywords

• nonn
• easy

Comments

This equation is used for worked examples in the Robertson paper.

(1/8)*a(n)^2 -5 is a term of A152741. [Bruno Berselli, Aug 17 2013]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• John P. Robertson, Solving the generalized Pell equation x^2 - Dy^2 = N
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,1298,0,0,0,-1).

Crossrefs

Cf. A228204.

Programs

Magma

I:=[12,40,220,768,14808,51700,285520,996852]; [n le 8 select I[n] else 1298*Self(n-4)-Self(n-8): n in [1..30]]; // Vincenzo Librandi, Aug 17 2013

Mathematica

CoefficientList[Series[-4 (x - 1) (3 x^6 + 13 x^5 + 68 x^4 + 260 x^3 + 68 x^2 + 13 x+3) / ((x^4 - 36 x^2 - 1) (x^4 + 36 x^2 - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 17 2013 *)
LinearRecurrence[{0,0,0,1298,0,0,0,-1},{12,40,220,768,14808,51700,285520,996852},30] (* Harvey P. Dale, Apr 22 2024 *)

PARI

Vec(-4*x*(x-1)*(3*x^6+13*x^5+68*x^4+260*x^3+68*x^2+13*x+3)/((x^4-36*x^2-1)*(x^4+36*x^2-1)) + O(x^100))

Formula

G.f.: -4*x*(x-1)*(3*x^6+13*x^5+68*x^4+260*x^3+68*x^2+13*x+3) / ((x^4-36*x^2-1)*(x^4+36*x^2-1)).

a(n) = 1298*a(n-4)-a(n-8).