A155136 Integers k such that k + 28 is a square.

-28, -27, -24, -19, -12, -3, 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
• easy

Comments

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

Agrees with A155135 except for omission of zero after a(6) = 3.

Examples

			For k = -19, k + 28 = 9 = 3^2 is a square.
For k = -3, k + 28 = 25 = 5^2 is a square.
For k = 21, k + 28 = 49 = 7^2 is a square.
		

Links

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

Crossrefs

Cf. A153642, A155135, A155137, A155138.

Programs

Magma

[ n: n in [ -30..2500] | IsSquare(n+28) ];

Mathematica

Range[0,50]^2-28 (* or  *) LinearRecurrence[{3,-3,1},{-28,-27,-24},50] (* Harvey P. Dale, May 15 2023 *)

PARI

a(n)=n^2-2*n-27 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = n^2 - 2*n - 27.

G.f.: -(4-3*x)*(7-9*x)/(1-x)^3.

From Elmo R. Oliveira, Oct 31 2024: (Start)

E.g.f.: exp(x)*(x^2 - x - 27) + 27.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)