A307168 First class of all proper positive solutions x1(n) = a(n) of the Pell equation x^2 - 7*y^2 = 9.

11, 172, 2741, 43684, 696203, 11095564, 176832821, 2818229572, 44914840331, 715819215724, 11408192611253, 181815262564324, 2897636008417931, 46180360872122572, 735988137945543221, 11729629846256568964, 186938089402159560203, 2979279800588296394284, 47481538720010582748341

Offset

1,1

Comments

The corresponding solutions y1(n) are given in A307169.

The (generalized) Pell equation x^2 -7*y^2 = 9 has two proper classes of positive solutions (x1(n), y1(n)) and (x2(n), y2(n), for n >= 1, where x2 = A307172, and y2 = A307173.

The improper class of nonnegative solutions is given by (xi(n) = 3*X(n), yi(n) = 3*Y(n)), with the nonnegative solutions of the Pell equation X^2 - 7*Y^2 = +1, given by X(n) = A001081(n) and Y(n) = A001080(n), for n >= 0.

The proper positive solutions (x1(n), y1(n)) are given in matrix notation by -R(0)*R(2)*Auto(n)*R^{-1}(6)*(1, 0)^T (T for transposed) with the R-matrix R(t) = Matrix([[0, -1],[1, t]]) and its inverse R^{-1}(t) = Matrix([t, 1],[-1, 0]) and the automorphic matrix Auto = Matrix([2, 9],[3, 14]). The matrix power Auto^n can be given in terms of Chebyshev S-polynomials S(n, x=16) from A077412 as Auto^n = Matrix([S(n, 16) - 14*S(n-1, 16), 9*S(n-1, 16)],[3*S(n-1, 16), S(n, 16) - 2*S(n-1, 16)]).

This results from the reduced principal binary quadratic form F_p = [1, 4, -3] of the non-reduced Pell form FPell = [1, 0, -7], and the primitive representative parallel form FPara1 = [9, 8, 1] for discriminant 4*7 = 28 and the representation of 9. These forms are then connected via equivalence transformations using R(t) matrices.

Links

Table of n, a(n) for n=1..19.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (16,-1).

Formula

a(n) = 4*S(n, 16) - 53*S(n-1, 16), for n >= 1, with S(n, 16) = A077412(n).

a(n) = sqrt(9 + 7*A307169(n)), n >= 1.

G.f.: x*(11 - 4*x)/(1 - 16*x + x^2).

Example

The solutions (x1(n), y1(n)) begin: (11, 4), (172, 65), (2741, 1036), (43684, 16511), (696203, 263140), (11095564, 4193729), (176832821, 66836524), (2818229572, 1065190655), (44914840331, 16976213956), ...
The solutions (x2(n), y2(n)) begin: (4, 1), (53, 20), (844, 319), (13451, 5084), (214372, 81025), (3416501, 1291316), (54449644, 20580031), (867777803, 327989180), (13829995204, 5227246849), ...
The improper solutions (xi(n), yi(n)) begin: (3, 0), (24, 9), (381, 144), (6072, 2295), (96771, 36576), (1542264, 582921), (24579453, 9290160), (391728984, 148059639), (6243084291, 2359664064), ...

Crossrefs

Cf. A001080, A001081, A077412, A307169, A307172, A307173.

Sequence in context: A025758 A243677 A374302 * A141955 A133243 A230604

Adjacent sequences: A307165 A307166 A307167 * A307169 A307170 A307171

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 27 2019

Status

approved