A281584 Solutions x to the negative Pell equation x^2 - 15*y^2 = -11 with x, y > 0.

2, 7, 23, 58, 182, 457, 1433, 3598, 11282, 28327, 88823, 223018, 699302, 1755817, 5505593, 13823518, 43345442, 108832327, 341257943, 856835098, 2686718102, 6745848457, 21152486873, 53109952558, 166533176882, 418133772007, 1311112928183, 3291960223498

Offset

1,1

Comments

The corresponding values of y are in A237262.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,8,0,-1).

Formula

G.f.: x*(1 + x)*(2 + x)*(1 + 2*x) / (1 - 8*x^2 + x^4).

a(n) = 8*a(n-2) - a(n-4) for n>4.

From Bruno Berselli, Jan 25 2017: (Start)

a(n) = (11*a(n-1) - 4*a(n-2))/3 if n is odd, a(n) = (11*a(n-1) - 3*a(n-2))/4 if n is even (see also R. J. Mathar in A237262).

a(n)*a(n-3) - a(n-1)*a(n-2) = -15*(7-(-1)^n)/2, with n>3. Example: for n=8, a(8)*a(5) - a(7)*a(6) = 3598*182 - 1433*457 = -15*3. (End)

Example

7 is in the sequence because (x, y) = (7, 2) is a solution to x^2 - 15*y^2 = -11.

Mathematica

LinearRecurrence[{0, 8, 0, -1}, {2, 7, 23, 58}, 30] (* Bruno Berselli, Jan 25 2017 *)

Prog

(PARI) Vec(x*(1 + x)*(2 + x)*(1 + 2*x) / (1 - 8*x^2 + x^4) + O(x^30))
(Magma) [n le 2 select 5*n-3 else IsOdd(n) select (11*Self(n-1)-4*Self(n-2))/3 else (11*Self(n-1)-3*Self(n-2))/4: n in [1..30]]; // Bruno Berselli, Jan 25 2017

Crossrefs

Cf. A237262.

Sequence in context: A041159 A210109 A034546 * A230315 A228629 A010748

Adjacent sequences: A281581 A281582 A281583 * A281585 A281586 A281587

Keyword

nonn,easy

Author

Colin Barker, Jan 25 2017

Status

approved