A176370 - OEIS

%I #14 Sep 08 2022 08:45:52

%S 1,65,8449,1098305,142771201,18559157825,2412547746049,

%T 313612647828545,40767231669964801,5299426504447595585,

%U 688884678346517461249,89549708758542822366785,11640773253932220390220801

%N x-values in the solution to x^2 - 66*y^2 = 1.

%C The corresponding values of y of this Pell equation are in A176372.

%H Vincenzo Librandi, <a href="/A176370/b176370.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (130,-1).

%F a(n) = 130*a(n-1) - a(n-2) with a(1)=1, a(2)=65.

%F G.f.: x*(1-65*x)/(1-130*x+x^2).

%p seq(coeff(series(x*(1-65*x)/(1-130*x+x^2), x, n+1), x, n), n = 1..15); # _G. C. Greubel_, Dec 08 2019

%t LinearRecurrence[{130,-1},{1,65},30]

%o (Magma) I:=[1, 65]; [n le 2 select I[n] else 130*Self(n-1)-Self(n-2): n in [1..20]];

%o (PARI) my(x='x+O('x^15)); Vec(x*(1-65*x)/(1-130*x+x^2)) \\ _G. C. Greubel_, Dec 08 2019

%o (Sage)

%o def A176368_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( x*(1-65*x)/(1-130*x+x^2) ).list()

%o a=A176368_list(15); a[1:] # _G. C. Greubel_, Dec 08 2019

%o (GAP) a:=[1,65];; for n in [3..15] do a[n]:=130*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 08 2019

%Y Cf. A176372.

%K nonn,easy

%O 1,2

%A _Vincenzo Librandi_, Apr 16 2010