%I #17 Oct 26 2024 04:57:46

%S 84,336,756,1344,2100,3024,4116,5376,6804,8400,10164,12096,14196,

%T 16464,18900,21504,24276,27216,30324,33600,37044,40656,44436,48384,

%U 52500,56784,61236,65856,70644,75600,80724,86016,91476,97104,102900,108864,114996,121296

%N Area A of the triangles such that A, the sides and one of the altitudes are four consecutive integers of an arithmetic progression d.

%C See the first link for the comments. We use Heron's Formula for a triangle: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.The sides and one of the altitudes are of the form q+d, q, q-d, q-2d.

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/Altitude.html">Altitude</a>

%F Conjecture: a(n) = 84*n^2. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: 84*x*(1+x)/(1-x)^3. - _Colin Barker_, Apr 19 2012

%e 84 is in the sequence because (a, b, c, h) = (15, 14, 13, 12) => A = sqrt(21*(21-15)*(21-14)*(21-13)) = sqrt(7056) = 84 but A = (1/2)*h*b = (1/2)*12*14 = 84.

%p with(numtheory):T:=array(1..1000):k:=0:nn:=800:for q from 1 to nn do: for d from 1 to nn do: a:=q+d:b:=q:c:=q-d:h1:=q-2*d:p:=(a+b+c)/2:s:=p*(p-a)*(p-b)*(p-c):if s>0 then s1:=sqrt(s): h11:=2*s1/a: h22:=2*s1/b:h33:=2*s1/c:if s1=floor(s1) and (h1=h11 or h1=h22 or h1=h33) then k:=k+1:T[k]:=s1:else fi:fi:od:od: L := [seq(T[i],i=1..k)]:L1:=convert(T,set):A:=sort(L1, `<`): print(A):

%Y Cf. A188158, A210643.

%K nonn,changed

%O 1,1

%A _Michel Lagneau_, Mar 26 2012