A155135 Integers k such that k^3+28*k^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

1,1

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

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

a(n) = n^2 - 4*n - 24 for n > 7. - Charles R Greathouse IV, May 26 2026

Example

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

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 *)
(* Alternative: *)
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 *)

Prog

(Magma) [ n: n in [ -30..2400] | IsSquare(n^3+28*n^2) ];
(PARI) a(n)=if(n>7, n^2-4*n-24, [-28, -27, -24, -19, -12, -3, 0][n]) \\ Charles R Greathouse IV, May 26 2026
(PARI) is(n) = issquare(28+n) || n == 0 \\ David A. Corneth, May 26 2026

Crossrefs

Cf. A155136, A155137, A153642.

Sequence in context: A196031 A196028 A291161 * A155136 A022984 A023470

Adjacent sequences: A155132 A155133 A155134 * A155136 A155137 A155138

Keyword

sign,easy,changed

Author

Klaus Brockhaus, Jan 21 2009

Status

approved