OFFSET

1,4

COMMENTS

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.

The sequence a(n) is the union of four subsequences A, B, C and D where:

A is the subsequence with areas 60, 210, 1716, 2730, 4080, 5814, 7980, 10626, ... where n is odd, and the corresponding sides are of the form (4k, 4k^2-1, 4k^2+1) with areas 2k(4k^2-1) for k = 2, 3, 6, 7, 8, 9, 11, ... These areas are in the sequence A069072 (areas of primitive Pythagorean triangles whose odd sides differ by 2).

B is the subsequence with areas 6, 30, 84, 330, 546, 2310, 4914, 6090, ... where n is even, and the corresponding sides are of the form (2k+1, 2k(k+1), 2k(k+1)+1) with areas k(k+1)(2k+1) for k = 1, 2, 3, 5, 6, 7, 10, 13, 14, ... These areas are in the sequence A055112 (Areas of Pythagorean triangles (a, b, c) with c = b+1.

C is the subsequence with areas 84, 198, 1380, 4176, ... where n is odd but the areas are not Pythagorean triangles.

D is the subsequence with areas 24, 60, 210, 924, 1248, 480, 1224, 1938, ... where n is even but the areas are not Pythagorean triangles.

The triangles with the same areas are not unique; for example:

(8, 15, 17) and (6, 25, 29) => A = 60; the first is a Pythagorean triangle, the second is not.

(12, 35, 37) and (7, 65, 68) => A = 210; the first is a Pythagorean triangle, the second is not.

The following table gives the first values (n, A, p, a, b, c) where A is the area of the triangles, p is the perimeter and a, b, c are the sides.

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

| n | A | p | a | b | c |

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

| 4 | 6 | 12 = 4*3 | 3 | 4 | 5 |

| 5 | 60 | 40 = 5*8 | 8 | 15 | 17 |

| 6 | 30 | 30 = 6*5 | 5 | 12 | 13 |

| 7 | 210 | 84 = 7*12 | 12 | 35 | 37 |

| 8 | 24 | 32 = 8*4 | 4 | 13 | 15 |

| 9 | 84 | 72 = 9*8 | 8 | 29 | 35 |

| 10 | 60 | 60 = 10*6 | 6 | 25 | 29 |

| 11 | 198 | 132 = 11*12 | 12 | 55 | 65 |

| 12 | 330 | 132 = 12*11 | 11 | 60 | 61 |

| 13 | 1716 | 312 = 13*24 | 24 | 143 | 145 |

| 14 | 546 | 182 = 14*13 | 13 | 84 | 85 |

| 15 | 2730 | 420 = 15*28 | 28 | 195 | 197 |

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

MAPLE

with(numtheory):nn:=600:for n from 4 to 50 do: ii:=0:for a from 1

to nn while(ii=0) do: for b from a to nn while(ii=0) do: for c from b to nn while(ii=0) do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x0:= sqrt(x):else fi:if x0=floor(x0) and 2*p=n*a then ii:=1:printf ( "%d %d %d %d %d \n", n, x0, a, b, c):else fi:od:od:od:od:

MATHEMATICA

nn=600; lst={}; Do[k=0; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0<area2&&k==0&&IntegerQ[Sqrt[area2]]&&2*s==n*Min[a, b, c], AppendTo[lst, Sqrt[area2]]; k=1]], {a, nn}, {b, a}, {c, b}], {n, 4, 45}]; Union[lst]

CROSSREFS

KEYWORD

nonn

AUTHOR

Michel Lagneau, Feb 09 2014

STATUS

approved