A155137 -id:A155137 - cp's OEIS frontend

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

Klaus Brockhaus, Jan 21 2009

sign

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.

			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.

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

Cf. A155136, A155137, A153642.

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

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 *)

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

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

Original entry on oeis.org

-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

Klaus Brockhaus, Jan 21 2009

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.

			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.

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

Cf. A153642, A155135, A155137, A155138.

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

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

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

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)

A155138 a(n) = nonnegative value y such that (A155136(n), y) is a solution to the Diophantine equation x^3+28*x^2 = y^2.

Original entry on oeis.org

0, 27, 48, 57, 48, 15, 48, 147, 288, 477, 720, 1023, 1392, 1833, 2352, 2955, 3648, 4437, 5328, 6327, 7440, 8673, 10032, 11523, 13152, 14925, 16848, 18927, 21168, 23577, 26160, 28923, 31872, 35013, 38352, 41895, 45648, 49617, 53808, 58227, 62880

Offset: 1

Klaus Brockhaus, Jan 21 2009

nonn

Agrees with A155137 except for omission of zero after a(6) = 15.

			(A155136(4), a(4)) = (-19, 57) is a solution: (-19)^3+28*(-19)^2 = -6859+10108 = 3249 = 57^2.
(A155136(8), a(8)) = (21, 147) is a solution: 21^3+28*21^2 = 9261+12348 = 21609 = 147^2.

Cf. A155136, A155137, A153642.

[ Abs((n-1)^3-28*(n-1)): n in [1..41] ];

Abs[#^3-28#]&/@Range[0,40] (* Harvey P. Dale, Aug 30 2016 *)

a(n) = Abs((n-1)^3-28*(n-1)).

G.f.: 3*x*(9-20*x+9*x^2+32*x^5-30*x^6-8*x^7+10*x^8)/(1-x)^4.

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

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

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A155138 a(n) = nonnegative value y such that (A155136(n), y) is a solution to the Diophantine equation x^3+28*x^2 = y^2.

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