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A237576
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Smallest integer areas of integer-sided triangles such that the perimeter equals n times the smallest side.
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0
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0, 0, 0, 6, 60, 30, 210, 24, 84, 60, 198, 330, 1716, 546, 2730, 252, 4080, 36, 5814, 210, 7980, 2310, 10626, 924, 1380, 1248, 90, 4914, 4176, 6090, 26970, 480, 32736, 1224, 39270, 1938, 46620, 2394, 54834, 4560, 63960, 4620, 74046, 19866, 85140, 22770, 97290
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OFFSET
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1,4
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COMMENTS
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The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.
The sequence a(n) is the union of four subsequences A, B, C and D where:
A is the subsequence with areas 60, 210, 1716, 2730, 4080, 5814, 7980, 10626, ... where n is odd, and the corresponding sides are of the form (4k, 4k^2-1, 4k^2+1) with areas 2k(4k^2-1) for k = 2, 3, 6, 7, 8, 9, 11, ... These areas are in the sequence A069072 (areas of primitive Pythagorean triangles whose odd sides differ by 2).
B is the subsequence with areas 6, 30, 84, 330, 546, 2310, 4914, 6090, ... where n is even, and the corresponding sides are of the form (2k+1, 2k(k+1), 2k(k+1)+1) with areas k(k+1)(2k+1) for k = 1, 2, 3, 5, 6, 7, 10, 13, 14, ... These areas are in the sequence A055112 (Areas of Pythagorean triangles (a, b, c) with c = b+1.
C is the subsequence with areas 84, 198, 1380, 4176, ... where n is odd but the areas are not Pythagorean triangles.
D is the subsequence with areas 24, 60, 210, 924, 1248, 480, 1224, 1938, ... where n is even but the areas are not Pythagorean triangles.
The triangles with the same areas are not unique; for example:
(8, 15, 17) and (6, 25, 29) => A = 60; the first is a Pythagorean triangle, the second is not.
(12, 35, 37) and (7, 65, 68) => A = 210; the first is a Pythagorean triangle, the second is not.
The following table gives the first values (n, A, p, a, b, c) where A is the area of the triangles, p is the perimeter and a, b, c are the sides.
+----+------+-------------+----+-----+-----+
| n | A | p | a | b | c |
+----+------+-------------+----+-----+-----+
| 4 | 6 | 12 = 4*3 | 3 | 4 | 5 |
| 5 | 60 | 40 = 5*8 | 8 | 15 | 17 |
| 6 | 30 | 30 = 6*5 | 5 | 12 | 13 |
| 7 | 210 | 84 = 7*12 | 12 | 35 | 37 |
| 8 | 24 | 32 = 8*4 | 4 | 13 | 15 |
| 9 | 84 | 72 = 9*8 | 8 | 29 | 35 |
| 10 | 60 | 60 = 10*6 | 6 | 25 | 29 |
| 11 | 198 | 132 = 11*12 | 12 | 55 | 65 |
| 12 | 330 | 132 = 12*11 | 11 | 60 | 61 |
| 13 | 1716 | 312 = 13*24 | 24 | 143 | 145 |
| 14 | 546 | 182 = 14*13 | 13 | 84 | 85 |
| 15 | 2730 | 420 = 15*28 | 28 | 195 | 197 |
+----+------+-------------+----+-----+-----+
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LINKS
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MAPLE
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with(numtheory):nn:=600:for n from 4 to 50 do: ii:=0:for a from 1
to nn while(ii=0) do: for b from a to nn while(ii=0) do: for c from b to nn while(ii=0) do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x0:= sqrt(x):else fi:if x0=floor(x0) and 2*p=n*a then ii:=1:printf ( "%d %d %d %d %d \n", n, x0, a, b, c):else fi:od:od:od:od:
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MATHEMATICA
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nn=600; lst={}; Do[k=0; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0<area2&&k==0&&IntegerQ[Sqrt[area2]]&&2*s==n*Min[a, b, c], AppendTo[lst, Sqrt[area2]]; k=1]], {a, nn}, {b, a}, {c, b}], {n, 4, 45}]; Union[lst]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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