A237576 Smallest integer areas of integer-sided triangles such that the perimeter equals n times the smallest side.

0, 0, 0, 6, 60, 30, 210, 24, 84, 60, 198, 330, 1716, 546, 2730, 252, 4080, 36, 5814, 210, 7980, 2310, 10626, 924, 1380, 1248, 90, 4914, 4176, 6090, 26970, 480, 32736, 1224, 39270, 1938, 46620, 2394, 54834, 4560, 63960, 4620, 74046, 19866, 85140, 22770, 97290

Offset: 1

Author

Michel Lagneau, Feb 09 2014

Keywords

• nonn

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 |

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

Crossrefs

Cf. A188158.

Programs

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