|
%I #17 Mar 11 2024 23:10:38
%S 11,172,2741,43684,696203,11095564,176832821,2818229572,44914840331,
%T 715819215724,11408192611253,181815262564324,2897636008417931,
%U 46180360872122572,735988137945543221,11729629846256568964,186938089402159560203,2979279800588296394284,47481538720010582748341
%N First class of all proper positive solutions x1(n) = a(n) of the Pell equation x^2 - 7*y^2 = 9.
%C The corresponding solutions y1(n) are given in A307169.
%C 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.
%C 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.
%C 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)]).
%C 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.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).
%F a(n) = 4*S(n, 16) - 53*S(n-1, 16), for n >= 1, with S(n, 16) = A077412(n).
%F a(n) = sqrt(9 + 7*A307169(n)), n >= 1.
%F G.f.: x*(11 - 4*x)/(1 - 16*x + x^2).
%e 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), ...
%e 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), ...
%e 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), ...
%Y Cf. A001080, A001081, A077412, A307169, A307172, A307173.
%K nonn,easy
%O 1,1
%A _Wolfdieter Lang_, Mar 27 2019
|
