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A232461 Integer areas of integer-sided triangles where two sides are of square length. 1
120, 168, 300, 360, 1920, 2016, 2688, 4680, 4800, 5760, 9720, 10140, 13608, 14280, 18720, 19080, 23256, 24300, 29160, 30720, 32760, 34440, 34680, 38640, 42120, 43008, 57720, 74880, 75000, 76800, 92160, 94080, 105000, 128700, 162240, 177072, 187500, 217728 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Subset of A188158.

The areas of the triangles (a,b,c) are given by Heron's formula, A = sqrt(s(s-a)(s-b)(s-c)), where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.

The areas A of the primitive triangles of sides (a,b,c) are 120, 168, 300, 360, 4680, ...

The areas of the nonprimitive triangle of sides (a*p^2, b*p^2, c*p^2) are in the sequence with the value A*p^4.

It is possible to find integer-sided triangles having two square sides, for example:

a(2) = 168 with sides (25,25,48) and (14,25,25);

a(3) = 300 with sides (25,25,30) and (25,25,40);

a(14) = 14280 with sides (169,169,238), (169,169,240), (100,289,291).

The following table gives the first values (A, a, b, c):

+-------+-----+-----+-----+

|   A   |  a  |  b  |  c  |

+-------+-----+-----+-----+

|   120 |  16 |  25 |  39 |

|   168 |  14 |  25 |  25 |

|   300 |  25 |  25 |  40 |

|   360 |  25 |  29 |  36 |

|  1920 |  64 | 100 | 156 |

|  2016 |  64 | 225 | 287 |

|  2688 | 100 | 100 | 192 |

|  4680 |  74 | 169 | 225 |

|  4800 | 100 | 100 | 160 |

|  5760 | 100 | 116 | 144 |

|  9720 | 144 | 225 | 351 |

| 10140 | 169 | 169 | 312 |

| 13608 | 225 | 225 | 432 |

+-------+-----+-----+-----+

LINKS

Table of n, a(n) for n=1..38.

J. Peng, Y. Zhang, Heron triangles with figurate number sides, Acta Mathematica Hungarica (2019) 1-11.

EXAMPLE

120 is in the sequence because the triangle (4^2, 5^2, 39) has semiperimeter s = (16+25+39)/2 = 40, and A = sqrt(40*(40-16)*(40-25)*(40-39)) = 120.

MATHEMATICA

nn=1000; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0<area2&&((IntegerQ[Sqrt[a]]&&IntegerQ[Sqrt[b]])||(IntegerQ[Sqrt[a]]&&IntegerQ[Sqrt[c]])||(IntegerQ[Sqrt[b]]&&IntegerQ[Sqrt[c]]))&&IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

sqr[n_]:=n^2; area[a_, b_, c_] := Module[{s = (a + b + c)/2, a2}, a2 = s (s - a) (s - b) (s - c); If[a2 < 0, 0, Sqrt[a2]]]; goodQ[a_, b_, c_] := Module[{ar = area[a, b, c]}, ar > 0 && IntegerQ[ar]]; nn = 80; t = {}; ps = sqr[Range[2, nn]]; mx = 3*ps[[-1]]; Do[If[p <= q && goodQ[p, q, e], aa = area[p, q, e]; If[aa <= mx, AppendTo[t, aa]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* program from T. D. Noe adapted for this sequence - see A229746 *)

CROSSREFS

Cf. A000290, A188158.

Sequence in context: A030634 A272594 A189975 * A090782 A023197 A204828

Adjacent sequences:  A232458 A232459 A232460 * A232462 A232463 A232464

KEYWORD

nonn

AUTHOR

Michel Lagneau, Nov 24 2013

STATUS

approved

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Last modified August 12 17:17 EDT 2020. Contains 336439 sequences. (Running on oeis4.)