|
%I #36 Jun 25 2022 21:52:05
%S 3,45,717,11427,182115,2902413,46256493,737201475,11748967107,
%T 187246272237,2984191388685,47559815946723,757972863758883,
%U 12080006004195405,192522123203367597,3068273965249686147,48899861320791610755,779329507167416085933,12420372253357865764173,197946626546558436140835
%N x-values of solutions to the Diophantine equation x^2 - 7*y^2 = 2.
%C A159678 gives the y-values of solutions to the Diophantine equation x^2 - 7*y^2 = 2.
%H G. C. Greubel, <a href="/A266698/b266698.txt">Table of n, a(n) for n = 1..750</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).
%F a(1)=3, a(2)=45, a(n) = 16*a(n-1) - a(n-2).
%F a(n) = A041008(4n-2). - _Robert Israel_, Jan 05 2016
%F From _R. J. Mathar_, Jan 12 2016: (Start)
%F G.f.: 3*x*(1-x) / ( 1-16*x+x^2 ).
%F a(n) = 3*A157456(n). (End)
%F From _G. C. Greubel_, Jun 25 2022: (Start)
%F a(n) = 3*(ChebyshevU(n-1, 8) - ChebyshevU(n-2, 8)).
%F E.g.f.: exp(8*x)*(3*cosh(3*sqrt(7)*x) - sqrt(7)*sinh(3*sqrt(7)*x)) - 3. (End)
%t LinearRecurrence[{16,-1}, {3, 45}, 20 ]
%o (PARI) lista(nn) = {print1(x = 3, ", "); print1(y = 45, ", "); for (n=2, nn, z = 16*y - x; print1(z, ", "); x = y; y = z;);} \\ _Michel Marcus_, Jan 05 2016
%o (Magma) [n: n in [1..2*10^7] | IsSquare((n^2-2)/7)]; // _Vincenzo Librandi_, Jan 06 2016
%o (SageMath) [3*(chebyshev_U(n-1, 8) - chebyshev_U(n-2, 8)) for n in (1..30)] # _G. C. Greubel_, Jun 25 2022
%Y Cf. A041008, A077412, A157456, A159678.
%K nonn,easy
%O 1,1
%A _Sture Sjöstedt_, Jan 03 2016
| 