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Numbers X such that 30*X^2-45 is a square.
2

%I #26 Apr 07 2024 14:39:12

%S 3,63,1383,30363,666603,14634903,321301263,7053992883,154866542163,

%T 3400009934703,74645352021303,1638797734533963,35978904807725883,

%U 789897108035435463,17341757471971854303,380728767275345359203

%N Numbers X such that 30*X^2-45 is a square.

%C Positive values of x (or y) satisfying x^2 - 22xy + y^2 + 180 = 0. - _Colin Barker_, Feb 19 2014

%H Vincenzo Librandi, <a href="/A133275/b133275.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 (22,-1).

%F a(n+2) = 22*a(n+1)-a(n).

%F a(n+1) = 11*a(n)+2*(30*a(n)^2-45)^0.5.

%F G.f.: -3*x*(-1+x)/(1-22*x+x^2). - _R. J. Mathar_, Nov 14 2007

%F a(n) = 3*A157014(n). - _Colin Barker_, Feb 19 2014

%t CoefficientList[Series[3 (1 - x)/(1 - 22 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 21 2014 *)

%o (Magma) I:=[3,63]; [n le 2 select I[n] else 22*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Feb 21 2014

%Y Cf. A157014.

%K nonn,easy

%O 1,1

%A _Richard Choulet_, Oct 16 2007

%E More terms from _Paolo P. Lava_, Aug 06 2008