A228204 y-values in the solution to x^2 - 13y^2 = 27.

3, 11, 61, 213, 4107, 14339, 79189, 276477, 5330883, 18612011, 102787261, 358866933, 6919482027, 24158375939, 133417785589, 465809002557, 8981482340163, 31357553356811, 173176182907261, 604619726452053, 11657957158049547, 40702080098764739

Offset: 1

Author

Colin Barker, Aug 16 2013

Keywords

• nonn
• easy

Comments

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

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. A228203.

Programs

Magma

I:=[3,11,61,213,4107,14339,79189,276477]; [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[(x + 1) (3 x^6 + 8 x^5 + 53 x^4 + 160 x^3 + 53 x^2 + 8 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},{3,11,61,213,4107,14339,79189,276477},30] (* Harvey P. Dale, Aug 26 2013 *)

PARI

Vec(x*(x+1)*(3*x^6+8*x^5+53*x^4+160*x^3+53*x^2+8*x+3)/((x^4-36*x^2-1)*(x^4+36*x^2-1)) + O(x^100))

Formula

G.f.: x*(x+1)*(3*x^6+8*x^5+53*x^4+160*x^3+53*x^2+8*x+3) / ((x^4-36*x^2-1)*(x^4+36*x^2-1)).

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