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A155135
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Integers n such that n^3+28*n^2 is a square.
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4
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-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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f.: -(28-57*x+27*x^2+8*x^6-11*x^7+3*x^9)/(1-x)^3.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(Magma) [ n: n in [ -30..2400] | IsSquare(n^3+28*n^2) ];
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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