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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).

%I A229926 #13 Mar 10 2017 21:43:32
%S A229926 6,12,24,48,96,192,384,768,1080,1080,3888,4320,15360,69120,69120,
%T A229926 248832,349920,349920,1259712,342144,7226112,10782720,17031168,
%U A229926 18095616,19226592,21660210,30270240,44706816,81544320,128798208
%N 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).
%C A229926 Subsequence of A188158.
%C A229926 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 A229926 a(n) = 6*2^n for n = 0, 1, 2,..., 7, and then this property disappears.
%C A229926 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 A229926 The following table gives the first values (n, A, a, b, c) where a <= b <= c are the integer sides of the triangles.
%C A229926 +----+------+-----+-----+-----+
%C A229926 |  n |    A |   a |   b |   c |
%C A229926 +----+------+-----+-----+-----+
%C A229926 |  0 |    6 |   3 |   4 |   5 |
%C A229926 |  1 |   12 |   5 |   5 |   6 |
%C A229926 |  2 |   24 |   6 |   8 |  10 |
%C A229926 |  3 |   48 |  10 |  10 |  12 |
%C A229926 |  4 |   96 |  12 |  16 |  20 |
%C A229926 |  5 |  192 |  20 |  20 |  24 |
%C A229926 |  6 |  384 |  24 |  32 |  40 |
%C A229926 |  7 |  768 |  40 |  40 |  48 |
%C A229926 |  8 | 1080 |  48 |  51 |  51 |
%C A229926 |  9 | 1080 |  51 |  51 |  90 |
%C A229926 | 10 | 3888 |  90 |  90 | 108 |
%C A229926 | 11 | 4320 | 108 | 116 | 208 |
%C A229926 +----+------+-----+-----+-----+
%p A229926 with(numtheory):nn:=15000:a:=5: printf ( "%d %d %d %d %d \n",1,6,3,4,a):
%p A229926     for n from 2 to 40 do:
%p A229926        ii:=0:
%p A229926       for b from a to nn while(ii=0) do:
%p A229926         for c from b to nn while(ii=0)  do:
%p A229926           p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c):
%p A229926           if x>0
%p A229926           then
%p A229926           x0:= sqrt(x):
%p A229926           else
%p A229926           fi:
%p A229926            if x0=floor(x0)
%p A229926            then
%p A229926            ii:=1:printf ( "%d %d %d %d %d \n",n,x0,a,b,c):
%p A229926            a:=max(b,c):
%p A229926            else
%p A229926            fi:
%p A229926           od:
%p A229926         od:
%p A229926        od:
%Y A229926 Cf. A188158.
%K A229926 nonn
%O A229926 0,1
%A A229926 _Michel Lagneau_, Oct 03 2013