A232461 Integer areas of integer-sided triangles where two sides are of square length.

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

Offset: 1

Author

Michel Lagneau, Nov 24 2013

Keywords

• nonn

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 |

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000290, A188158.

Programs

Mathematica

nn=1000;lst={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a) (s-b) (s-c);If[0 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 *)