A275794 One half of the y members of the positive proper solutions (x = x1(n), y = y1(n)) of the first class for the Pell equation x^2 - 2*y^2 = +7^2.

2, 15, 88, 513, 2990, 17427, 101572, 592005, 3450458, 20110743, 117214000, 683173257, 3981825542, 23207779995, 135264854428, 788381346573, 4595023225010, 26781758003487, 156095524795912, 909791390771985, 5302652819835998, 30906125528244003

Offset

0,1

Comments

See A275793(n) for the x1(n) members and details as well as a reference.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (6,-1).

Formula

a(n) = 15*S(n-1,6) - 2*S(n-2,6), with the Chebyshev polynomials S(n, 6) = A001109(n+1) for n >= -1, with S(-2, 6) = -1.

O.g.f: (2 + 3*x)/(1 - 6*x + x^2).

a(n) = 6*a(n-1) - a(n-2) for n >= 1, with a(-1) = -3 and a(0) = 2.

a(n) = (((3-2*sqrt(2))^n*(-9+4*sqrt(2))+(3+2*sqrt(2))^n*(9+4*sqrt(2)))) / (4*sqrt(2)). - Colin Barker, Sep 28 2016

Example

See A275793.

Mathematica

CoefficientList[Series[(2 + 3*x)/(1 - 6*x + x^2), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 29 2017 *)
LinearRecurrence[{6, -1}, {2, 15}, 30] (* Harvey P. Dale, May 30 2018 *)

Prog

(PARI) a(n) = round((((3-2*sqrt(2))^n*(-9+4*sqrt(2))+(3+2*sqrt(2))^n*(9+4*sqrt(2))))/(4*sqrt(2))) \\ Colin Barker, Sep 28 2016
(PARI) Vec((2+3*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Oct 02 2016

Crossrefs

Cf. A001109, A275793.

Sequence in context: A376135 A208974 A294196 * A037746 A037627 A192369

Adjacent sequences: A275791 A275792 A275793 * A275795 A275796 A275797

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 27 2016

Status

approved