A229926 - OEIS

%I #13 Mar 10 2017 21:43:32

%S 6,12,24,48,96,192,384,768,1080,1080,3888,4320,15360,69120,69120,

%T 248832,349920,349920,1259712,342144,7226112,10782720,17031168,

%U 18095616,19226592,21660210,30270240,44706816,81544320,128798208

%N 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).

%C Subsequence of A188158.

%C 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, ...}

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

%C 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.

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

%C +----+------+-----+-----+-----+

%C |  n |    A |   a |   b |   c |

%C +----+------+-----+-----+-----+

%C |  0 |    6 |   3 |   4 |   5 |

%C |  1 |   12 |   5 |   5 |   6 |

%C |  2 |   24 |   6 |   8 |  10 |

%C |  3 |   48 |  10 |  10 |  12 |

%C |  4 |   96 |  12 |  16 |  20 |

%C |  5 |  192 |  20 |  20 |  24 |

%C |  6 |  384 |  24 |  32 |  40 |

%C |  7 |  768 |  40 |  40 |  48 |

%C |  8 | 1080 |  48 |  51 |  51 |

%C |  9 | 1080 |  51 |  51 |  90 |

%C | 10 | 3888 |  90 |  90 | 108 |

%C | 11 | 4320 | 108 | 116 | 208 |

%C +----+------+-----+-----+-----+

%p with(numtheory):nn:=15000:a:=5: printf ( "%d %d %d %d %d \n",1,6,3,4,a):

%p     for n from 2 to 40 do:

%p        ii:=0:

%p       for b from a to nn while(ii=0) do:

%p         for c from b to nn while(ii=0)  do:

%p           p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c):

%p           if x>0

%p           then

%p           x0:= sqrt(x):

%p           else

%p           fi:

%p            if x0=floor(x0)

%p            then

%p            ii:=1:printf ( "%d %d %d %d %d \n",n,x0,a,b,c):

%p            a:=max(b,c):

%p            else

%p            fi:

%p           od:

%p         od:

%p        od:

%Y Cf. A188158.

%K nonn

%O 0,1

%A _Michel Lagneau_, Oct 03 2013