A281647 Solutions x to the negative Pell equation x^2 - 10*y^2 = -6 with x > y > 0.

2, 22, 98, 838, 3722, 31822, 141338, 1208398, 5367122, 45887302, 203809298, 1742509078, 7739386202, 66169457662, 293892866378, 2512696882078, 11160189536162, 95416312061302, 423793309507778, 3623307161447398, 16092985571759402, 137590255822939822

Offset

1,1

Comments

The corresponding values of y are in A221875.

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

Formula

G.f.: 2*x*(1 + x)*(1 + 10*x + x^2) / ((1 + 6*x - x^2)*(1 - 6*x - x^2)).

a(n) = 38*a(n-2) - a(n-4) for n>4.

a(n) = ((3-r)^n + (-3-r)^n*(-3+r) - 3*(-3+r)^n - r*(-3+r)^n + (3+r)^n)/2, where r=sqrt(10).

Example

22 is in the sequence because (x, y) = (22, 7) is a solution to x^2 - 10*y^2 = -6.

Mathematica

CoefficientList[ Series[(2 (1 + 11x + 11x^2 + x^3))/(1 - 38x^2 + x^4), {x, 0, 21}],  x] (* or *)
LinearRecurrence[{0, 38, 0, -1}, {2, 22, 98, 838}, 22] (* Robert G. Wilson v, Jan 26 2017 *)

Prog

(PARI) Vec(2*x*(1 + x)*(1 + 10*x + x^2) / ((1 + 6*x - x^2)*(1 - 6*x - x^2)) + O(x^30))

Crossrefs

Cf. A221875.

Sequence in context: A291915 A172229 A212894 * A344498 A281140 A105237

Adjacent sequences: A281644 A281645 A281646 * A281648 A281649 A281650

Keyword

nonn,easy

Author

Colin Barker, Jan 26 2017

Status

approved