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Numbers X such that there exists Y in N with X^2=111*Y^2+37.
1

%I #14 Jan 03 2024 23:46:13

%S 74,43586,25715666,15172199354,8951571903194,5281412250685106,

%T 3116024276332309346,1838449041623811829034,1084681818533772646820714,

%U 639960434485884237812392226,377575571664853166536664592626

%N Numbers X such that there exists Y in N with X^2=111*Y^2+37.

%H Vincenzo Librandi, <a href="/A145695/b145695.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 (590,-1).

%F a(n+2) = 590*a(n+1)-a(n).

%F G.f.: -74*x*(x-1) / (x^2-590*x+1). - _Colin Barker_, Oct 21 2014

%e a(1)=74 because the first relation is 74^2=111*7^2+37.

%t CoefficientList[Series[74 (1 - x)/(x^2 - 590 x + 1), {x, 0, 20}], x] (* _Vincenzo Librandi_, Oct 21 2014 *)

%o (PARI) Vec(-74*x*(x-1)/(x^2-590*x+1) + O(x^20)) \\ _Colin Barker_, Oct 21 2014

%o (Magma) I:=[74,43586]; [n le 2 select I[n] else 590*Self(n-1)-Self(n-2): n in [1..15]]; // _Vincenzo Librandi_, Oct 21 2014

%K easy,nonn

%O 1,1

%A _Richard Choulet_, Oct 16 2008

%E Editing and a(11) from _Colin Barker_, Oct 21 2014