A230491 Integer 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

Offset

1,1

Comments

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 *

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

References

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.

Links

Table of n, a(n) for n=1..49.

Eric W. Weisstein, MathWorld: Inradius

Formula

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

Example

84 is in the sequence because the area of triangle (13, 14, 15) is given by 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.

Mathematica

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]

Crossrefs

Cf. A000290, A188158, A228383.

Sequence in context: A186661 A186659 A196256 * A067249 A351980 A351178

Adjacent sequences: A230488 A230489 A230490 * A230492 A230493 A230494

Keyword

nonn

Author

Michel Lagneau, Oct 20 2013

Status

approved