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%I #9 Jan 10 2023 17:12:42
%S -28,-27,-24,-19,-12,-3,0,8,21,36,53,72,93,116,141,168,197,228,261,
%T 296,333,372,413,456,501,548,597,648,701,756,813,872,933,996,1061,
%U 1128,1197,1268,1341,1416,1493,1572,1653,1736,1821,1908,1997,2088,2181,2276,2373
%N Integers n such that n^3+28*n^2 is a square.
%C Values x of solutions (x, y) to the Diophantine equation x^3+28*x^2 = y^2. Corresponding values y are in A155137.
%C Agrees with A155136 except for insertion of zero after a(6) = 3.
%H Vincenzo Librandi, <a href="/A155135/b155135.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: -(28-57*x+27*x^2+8*x^6-11*x^7+3*x^9)/(1-x)^3.
%e For n = -19, n^3+28*n^2 = -6859+10108 = 3249 = 57^2 is a square.
%e For n = 0, n^3+28*n^2 = 0^3+28*0^2 = 0 = 0^2 is a square.
%e For n = 21; n^3+28*n^2 = 9261+12348 = 21609 = 147^2 is a square.
%t CoefficientList[Series[-(28-57*x+27*x^2+8*x^6-11*x^7+3*x^9)/(1-x)^3,{x,0,60}],x] (* _Vincenzo Librandi_, Feb 22 2012 *)
%t Select[Range[-30,2500],IntegerQ[Sqrt[#^3+28#^2]]&] (* or *) LinearRecurrence[ {3,-3,1},{-28,-27,-24,-19,-12,-3,0,8,21,36},60] (* _Harvey P. Dale_, Jan 10 2023 *)
%o (Magma) [ n: n in [ -30..2400] | IsSquare(n^3+28*n^2) ];
%Y Cf. A155136, A155137, A153642.
%K sign
%O 1,1
%A _Klaus Brockhaus_, Jan 21 2009
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