A228383 -id:A228383 - cp's OEIS frontend

A289155 Smallest area of triangle with integer sides and area = n times perimeter.

24, 84, 192, 336, 540, 756, 1134, 1344, 1710, 2100, 2640, 3000, 4056, 4116, 4680, 5376, 6936, 6804, 8664, 8400, 9240, 10164, 12696, 12000, 13500, 14196, 15390, 16296, 20184, 18720, 23064, 21504, 23232, 24276, 26040, 27000, 32856, 30324

Offset: 1

Zhining Yang, Jun 26 2017

nonn

			For n = 4, a(4)=336 means for the smallest triangle (a,b,c) = (26,28,30), the area is 336, which is 4 times the perimeter 84.

Ray Chandler, Table of n, a(n) for n = 1..5000 (first 300 terms from Zhining Yang)

Cf. A007237, A120572, A188158, A228383.

a(n) is the leading entry in row n of the triangle in A290451.

for(k=1, 50, n=0;A=10^9; d=4*k^2; e=3*d; for(b=1, sqrt(e), for (c=2*k, e/b, if(b*c>d&&c>=b, f = (b + c)*d / (b * c - d); if(f>=c, a=floor(f); if(a==f, n++; s=2*(a+b+c)*k;if(s

a(n) = A120572(2n). - Ray Chandler, Jul 27 2017

A289156 Largest area of triangles with integer sides and area = n times perimeter.

Original entry on oeis.org

60, 1224, 8436, 34320, 103020, 254040, 546084, 1060896, 1907100, 3224040, 5185620, 8004144, 11934156, 17276280, 24381060, 33652800, 45553404, 60606216, 79399860, 102592080, 130913580, 165171864, 206255076, 255135840, 312875100, 380625960, 459637524, 551258736

Offset: 1

Zhining Yang, Jun 26 2017

			For n = 4, a(4) = 34320 means for the largest triangles (a,b,c) = (66,4225,4289), the area is 34320 which is 4 times the perimeter 8580.

Ray Chandler, Table of n, a(n) for n = 1..5000 (first 100 terms from Zhining Yang)
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A007237, A120573, A188158, A228383, A289155.

Cf. A005900, A007900, A053755.

Table[4 n (2 n^2 + 1) (4 n^2 + 1), {n, 27}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {60, 1224, 8436, 34320, 103020, 254040}, 27] (* or *) Rest@ CoefficientList[Series[12 x (5 + 72 x + 166 x^2 + 72 x^3 + 5 x^4)/(1 - x)^6, {x, 0, 27}], x] (* Michael De Vlieger, Jul 03 2017 *)

Vec(12*x*(5 + 72*x + 166*x^2 + 72*x^3 + 5*x^4)/(1 - x)^6 + O(x^30)) \\ Colin Barker, Jun 28 2017

From Colin Barker, Jun 28 2017: (Start)
G.f.: 12*x*(5 + 72*x + 166*x^2 + 72*x^3 + 5*x^4)/(1 - x)^6.
a(n) = 4*n*(2*n^2 + 1)*(4*n^2 + 1).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6. (End)
a(n) = A120573(2*n). - Ray Chandler, Jul 27 2017
From Elmo R. Oliveira, Sep 01 2025: (Start)
E.g.f.: 4*exp(x)*x*(15 + 138*x + 206*x^2 + 80*x^3 + 8*x^4).
a(n) = 12*A005900(n)*A053755(n) = A053755(n)*A007900(n)/2. (End)

A230195 Integer areas A of the triangles such that A and the sides are integers, and the length of the inradius is a prime number.

Original entry on oeis.org

24, 30, 36, 42, 48, 54, 60, 66, 84, 96, 114, 120, 126, 150, 156, 198, 210, 270, 294, 330, 336, 390, 420, 462, 504, 510, 546, 570, 630, 714, 726, 756, 810, 840, 924, 930, 1008, 1014, 1056, 1134, 1386, 1428, 1554, 1638, 1680, 1716, 1734, 1848, 1890, 1950, 2016

Offset: 1

Michel Lagneau, Oct 11 2013

nonn

Subsequence of A228383.
The corresponding inradii r are 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 5, 7, 5, 7, 5, 7, 7, 7, 5, 7, 5, ...
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 inradius r is given by r = A/s.

			24 is in the sequence because for (a, b, c) = (6, 8, 10) => s =(6 + 8 + 10)/2 = 12; A = sqrt(12*(12-6)*(12-8)*(12-10)) = sqrt(576)= 24; r = A/s = 2 is prime.

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

Cf. A228383.

nn = 1500; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]] && PrimeQ[Sqrt[area2]/s], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

A230491 Integer areas of the integer-sided triangles such that the length of the inradius is a square.

Original entry on oeis.org

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

Michel Lagneau, Oct 20 2013

nonn

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  *
.........................

			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.

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, MathWorld: Inradius

Cf. A000290, A188158, A228383.

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]

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

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A289155 Smallest area of triangle with integer sides and area = n times perimeter.

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A289156 Largest area of triangles with integer sides and area = n times perimeter.

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A230195 Integer areas A of the triangles such that A and the sides are integers, and the length of the inradius is a prime number.

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A230491 Integer areas of the integer-sided triangles such that the length of the inradius is a square.

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