A229926 Integer areas of the integer-sided triangles T(n) defined by the property: a(0) = 6 ; for n > 0, a(n) is the area A where the smallest side of T(n) is the greatest side of T(n-1).

6, 12, 24, 48, 96, 192, 384, 768, 1080, 1080, 3888, 4320, 15360, 69120, 69120, 248832, 349920, 349920, 1259712, 342144, 7226112, 10782720, 17031168, 18095616, 19226592, 21660210, 30270240, 44706816, 81544320, 128798208

Offset: 0

Author

Michel Lagneau, Oct 03 2013

Keywords

• nonn

Comments

Subsequence of A188158.

The sequence of the common sides is {5, 6, 10, 12, 20, 24, 40, 48, 51, 90, 108, 208, 384, 408, 720, 864, 918, 1620, 1944, 3880, 4656, 6240, 6336, ...}

a(n) = 6*2^n for n = 0, 1, 2,..., 7, and then this property disappears.

The area is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where the semiperimeter s = (a + b + c)/2.

The following table gives the first values (n, A, a, b, c) where a <= b <= c are the integer sides of the triangles.

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

| n | A | a | b | c |

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

| 0 | 6 | 3 | 4 | 5 |

| 1 | 12 | 5 | 5 | 6 |

| 2 | 24 | 6 | 8 | 10 |

| 3 | 48 | 10 | 10 | 12 |

| 4 | 96 | 12 | 16 | 20 |

| 5 | 192 | 20 | 20 | 24 |

| 6 | 384 | 24 | 32 | 40 |

| 7 | 768 | 40 | 40 | 48 |

| 8 | 1080 | 48 | 51 | 51 |

| 9 | 1080 | 51 | 51 | 90 |

| 10 | 3888 | 90 | 90 | 108 |

| 11 | 4320 | 108 | 116 | 208 |

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

Crossrefs

Cf. A188158.

Programs

Maple

with(numtheory):nn:=15000:a:=5: printf ( "%d %d %d %d %d \n",1,6,3,4,a):
    for n from 2 to 40 do:
       ii:=0:
      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)
           then
           ii:=1:printf ( "%d %d %d %d %d \n",n,x0,a,b,c):
           a:=max(b,c):
           else
           fi:
          od:
        od:
       od: