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A228858 Let A(x,y,z) denote the integer area A of the triangle with integer side lengths (x,y,z). a(n) is the smallest area such that there exist exactly n distinct triangles (x_i, y_i, z_i) satisfying A(x_1, y_1, z_1) = A(x_2, y_2, z_2) = ... = A(x_n, y_n, z_n). 1
6, 12, 126, 60, 240, 210, 1080, 336, 1260, 924, 2016, 2640, 7980, 6930, 420, 2520, 840, 3696, 10080, 5460, 3360 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The area A of a triangle whose sides have lengths x, y, and z is given by A188158. The area is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2. The Maple program examines all triangles having longest side <= 600.
Conjecture: there exists an integer corresponding to the common area of exactly n distinct triangles (x_i, y_i, z_i) for i = 1,2,...,n such that A(x_1, y_1, z_1) = A(x_2, y_2, z_2) = ... = A(x_n, y_n, z_n) for n = 1, 2, ...
LINKS
Eric Weisstein's World of Mathematics, Triangle
EXAMPLE
a(6) = 210 because there exists exactly 6 triangles with the same area. We obtain:
(3,148,149) => A = sqrt(150*(150-3)*(150-148)*(150-149)) = 210;
(7,65,68) => A = sqrt(70*(70-7)*(70-65)*(70-68)) = 210;
(12,35,37) => A = sqrt(42*(42-12)*(42-35)*(42-37)) = 210;
(17,25,28) => A = sqrt(35*(35-17)*(35-25)*(35-28)) = 210;
(17,28,39) => A = sqrt(42*(42-17)*(42-28)*(42-39)) = 210;
(20,21,29) => A = sqrt(35*(35-20)*(35-21)*(35-29)) = 210.
MAPLE
with(numtheory):nn:=600:lst:={}:T:=array(1..30000):k:=0:for a from 1 to nn do: for b from a to nn do: for c from b to nn do:p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x1:=abs(x):s:=sqrt(x1) :else fi:if s=floor(s) then lst:=lst union {s}:k:=k+1:T[k]:=s:else fi:od:od:od:k1:=nops(lst):for n from 1 to 50 do:jj:=0:for i from 1 to k1 while(jj=0) do:ii:=0:for j from 1 to k while(jj=0) do:if lst[i]=T[j] then ii:=ii+1:else fi:od: if ii = n then jj:=1:printf ( "%d %d \n", n, lst[i]):else fi:od:od:
CROSSREFS
Sequence in context: A038515 A335735 A051586 * A102077 A102062 A164584
KEYWORD
nonn,hard
AUTHOR
Michel Lagneau, Sep 05 2013
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)