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Part of the positive proper solutions x of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (1, 5).
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%I #14 Jun 28 2019 07:15:36

%S 1,23,137,799,4657,27143,158201,922063,5374177,31322999,182563817,

%T 1064059903,6201795601,36146713703,210678486617,1227924205999,

%U 7156866749377,41713276290263,243122790992201,1417023469662943

%N Part of the positive proper solutions x of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (1, 5).

%C The corresponding y solutions are given in A254759.

%C The other part of the proper (sometimes called primitive) solutions are given in (A254757(n), A220414(n)) for n >= 1.

%C The improper positive solutions come from 7*(x(n), y(n)) with the positive proper solutions of the Pell equation x^2 - 2*y^2 = -1 given in (A001653(n-1), A002315(n)), for n >= 1.

%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.

%H Wolfdieter Lang, <a href="http://www.itp.kit.edu/~wl/BinQuadForm.html">Binary Quadratic Forms (indefinite case)</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, -1).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F a(n) = rational part of z(n), where z(n) = (1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 0.

%F G.f.: (1+17*x)/(1-6*x+x^2).

%F a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = -17 and a(0) = 1.

%F a(n) = S(n, 6) + 17*S(n-1, 6), n >= 0, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).

%e The first pairs of positive solutions of this part of the Pell equation x^2 - 2*y^2 = - 7^2 are: [1, 5], [23, 17], [137, 97], [799, 565], [4657, 3293], [27143, 19193], [158201, 111865], [922063, 651997], [5374177, 3800117], ...

%p with(orthopoly): a := n -> `if`(n=0,1, U(n,3)+17*U(n-1, 3)):

%p seq(a(n), n=0..19); # _Peter Luschny_, Feb 07 2015

%t LinearRecurrence[{6, -1}, {1, 23}, 20] (* _Jean-François Alcover_, Jun 28 2019 *)

%o (PARI) Vec((1+17*x)/(1-6*x+x^2) + O(x^30)) \\ _Michel Marcus_, Feb 08 2015

%Y Cf. A254759, A254757, A220414, A001653, A002315, A049310.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Feb 07 2015