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84 is in the sequence because the area of triangle (13, 14, 15) is given by the Heron’'s formula A = sqrt(21*(21-13)*(21-14)*(21-15))= 84 where the number 21 is the semiperimeter and the inradius is given by r = A/s = 84/21 = 4 is a square.

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84 is in the sequence because the area of triangle (13, 14, 15) is given by the Heron’s formula A = sqrt(21*(21-13)*(21-14)*(21-15))= 84 where the number 21 is the semi-perimeter semiperimeter and the inradius is given by r = A/s = 84/21 = 4 is a square.

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84 is in the sequence because the area of triangle (13, 14, 15) is given by the Heron’s formula A = sqrt(21*(21-13)*(21-14)*(21-15))= 84 where the number 21 is the semi -perimeter and the inradius is given by r = A/s = 84/21= 4 is a square.

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nn = 600; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 && IntegerQ[Sqrt[area2]] && IntegerQ[Sqrt[Sqrt[area2]/s]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Cf. A000290, A188158, A228383.

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allocated for Michel LagneauInteger areas of the integer-sided triangles such that the length of the inradius is a square.

6, 84, 96, 108, 120, 132, 144, 156, 168, 180, 240, 264, 300, 324, 396, 420, 432, 468, 486, 504, 540, 594, 630, 684, 720, 756, 864, 990, 1026, 1080, 1116, 1134, 1152, 1224, 1332, 1344, 1404, 1440, 1494, 1536, 1584, 1638, 1680, 1710, 1728, 1782, 1824, 1872, 1890

1,1

The primitive areas are 6, 84, 108, 120, 132, 144, 156, 168, ...

The non-primitive areas 16*a(n) are in the sequence because if r is the inradius corresponding to a(n), then 4*r is the inradius corresponding to 16*a(n).

The following table gives the first values (A, r, a, b, c) where A is the integer area, r the inradius and a, b, c are the integer sides of the triangle.

******************************

* A * r * a * b * c *

*******************************

* 6 * 1 * 3 * 4 * 5 *

* 84 * 4 * 13 * 14 * 15 *

* 96 * 4 * 12 * 16 * 20 *

* 108 * 4 * 15 * 15 * 24 *

* 120 * 4 * 10 * 24 * 26 *

* 132 * 4 * 11 * 25 * 30 *

* 144 * 4 * 18 * 20 * 34 *

* 156 * 4 * 15 * 26 * 37 *

* 168 * 4 * 10 * 35 * 39 *

* 180 * 4 * 9 * 40 * 41 *

* 240 * 4 * 12 * 50 * 58 *

* 264 * 4 * 33 * 34 * 65 *

* 300 * 4 * 25 * 51 * 74 *

* 324 * 4 * 9 * 75 * 78 *

* 396 * 4 * 11 * 90 * 97 *

* 420 * 4 * 21 * 85 * 104 *

* 432 * 9 * 30 * 30 * 36 *

* 468 * 9 * 25 * 39 * 40 *

.........................

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Inradius.html">MathWorld: Inradius</a>

Area A = sqrt(s*(s-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula) and inradius r = A/s.

84 is in the sequence because the area of triangle (13, 14, 15) is given by the Heron’s formula A = sqrt(21*(21-13)*(21-14)*(21-15))= 84 where the number 21 is the semi perimeter and the inradius is given by r = A/s = 84/21= 4 is a square.

nn=600; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s(s-a)(s-b)(s-c); If[0<area2&&IntegerQ[Sqrt[area2]]&&IntegerQ[Sqrt[Sqrt[area2]/s]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Cf. A000290, A188158,A228383.

allocated

nonn

Michel Lagneau, Oct 20 2013

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