A275795 The x members of the positive proper solutions (x = x2(n), y = y2(n)) of the second class for the Pell equation x^2 - 2*y^2 = +7^2.

11, 57, 331, 1929, 11243, 65529, 381931, 2226057, 12974411, 75620409, 440748043, 2568867849, 14972459051, 87265886457, 508622859691, 2964471271689, 17278204770443, 100704757350969, 586950339335371, 3420997278661257, 19939033332632171, 116213202717131769

Offset

0,1

Comments

For details and the Nagell reference see A275793.

The y2(n) members are given in A275795(n).

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) = 57*S(n-1,6) - 11*S(n-2,6) with the Chebyshev polynomials S(n, 6) = A001109(n+1), n >= -1, with S(-2, 6) = -1.

O.g.f.: (11 - 9*x)/(1 - 6*x + x^2).

a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = 9 and a(0) = 11.

a(n) = (((3-2*sqrt(2))^n*(-12+11*sqrt(2))+(3+2*sqrt(2))^n*(12+11*sqrt(2)))) / (2*sqrt(2)). - Colin Barker, Sep 28 2016

Mathematica

LinearRecurrence[{6, -1}, {11, 57}, 30] (* Harvey P. Dale, Sep 01 2022 *)

Prog

(PARI) a(n) = round((((3-2*sqrt(2))^n*(-12+11*sqrt(2))+(3+2*sqrt(2))^n*(12+11*sqrt(2)))) / (2*sqrt(2))) \\ Colin Barker, Sep 28 2016
(PARI) Vec((11-9*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Oct 09 2016

Crossrefs

Cf. A275793, A275794, A275795.

Sequence in context: A187693 A200529 A289255 * A014470 A285448 A048366

Adjacent sequences: A275792 A275793 A275794 * A275796 A275797 A275798

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 27 2016

Status

approved