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A210207 Area A of the non-right triangles such that A, the sides, and the circumradius are integers. 3
168, 432, 480, 624, 672, 768, 1320, 1512, 1536, 1560, 1680, 1728, 1848, 1920, 2040, 2304, 2376, 2496, 2520, 2688, 2856, 3024, 3072, 3240, 3696, 3720, 3840, 3864, 3888, 4104, 4200, 4320, 4536, 5280, 5376, 5616, 5712, 6000, 6048, 6144, 6240, 6552, 6720, 6912 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A103251 gives the areas of right triangles with the same property (the area, the sides, and the circumradius are integers). Thus the intersection of this sequence with A103251 will give the areas of 2 families of triangles with the same property: one family of right triangles and one family of non-right triangles.

For example a(3) = A103251(8) = 480 generates two triangles whose sides are

(a,b,c) = (32, 50, 78) = > A = 480, R = 65, and 32^2 + 50^2 is no square;

(a,b,c) = (20, 48, 52) = > A = 480, R = 26, and 20^2 + 48^2  = 52^2 is square.

{a(n) intersection A103251} = {480, 1320, 1536, 1920, 2520, 3024, 3696, 3840, ...}

LINKS

Table of n, a(n) for n=1..44.

Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.

Eric W. Weisstein, MathWorld: Circumradius

FORMULA

Area A = sqrt(s*(s-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula);

Circumradius R = a*b*c/4A.

EXAMPLE

168 is in the sequence because, for (a,b,c) = (14,30,40), A = sqrt(42*(42-14)*(42-30)*(42-40)) = 168, and 14^2 + 30^2 is no square.

MAPLE

T:=array(1..4000):nn:=400: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): u:=a^2+b^2:if  x>0 then x1:=sqrt(x) : y:=a*b*c/(4*x1):

else fi:if x1=floor(x1) and y = floor(y) and u <> c^2 then k:=k+1:T[k]:=x1:else fi:od:od:od: L := [seq(T[i], i=1..k)]:L1:=convert(T, set):A:=sort(L1, `<`): print(A):

MATHEMATICA

nn=400; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0 < area2 && a^2 != b^2+c^2 && IntegerQ[Sqrt[area2]] && IntegerQ[a*b*c/(4*Sqrt[area2])], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

CROSSREFS

Cf. A103251, A208984, A188158.

Sequence in context: A105915 A158219 A273771 * A303082 A247721 A027679

Adjacent sequences:  A210204 A210205 A210206 * A210208 A210209 A210210

KEYWORD

nonn

AUTHOR

Michel Lagneau, Mar 18 2012

STATUS

approved

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Last modified August 6 15:39 EDT 2020. Contains 336253 sequences. (Running on oeis4.)