A155135 Integers n such that n^3+28*n^2 is a square.

-28, -27, -24, -19, -12, -3, 0, 8, 21, 36, 53, 72, 93, 116, 141, 168, 197, 228, 261, 296, 333, 372, 413, 456, 501, 548, 597, 648, 701, 756, 813, 872, 933, 996, 1061, 1128, 1197, 1268, 1341, 1416, 1493, 1572, 1653, 1736, 1821, 1908, 1997, 2088, 2181, 2276, 2373

Offset: 1

Author

Klaus Brockhaus, Jan 21 2009

Keywords

• sign

Comments

Values x of solutions (x, y) to the Diophantine equation x^3+28*x^2 = y^2. Corresponding values y are in A155137.

Agrees with A155136 except for insertion of zero after a(6) = 3.

Examples

			For n = -19, n^3+28*n^2 = -6859+10108 = 3249 = 57^2 is a square.
For n = 0, n^3+28*n^2 = 0^3+28*0^2 = 0 = 0^2 is a square.
For n = 21; n^3+28*n^2 = 9261+12348 = 21609 = 147^2 is a square.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A155136, A155137, A153642.

Programs

Magma

[ n: n in [ -30..2400] | IsSquare(n^3+28*n^2) ];

Mathematica

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 *)
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 *)

Formula

G.f.: -(28-57*x+27*x^2+8*x^6-11*x^7+3*x^9)/(1-x)^3.