A120062 -id:A120062 - cp's OEIS frontend

A386980 Number of acute Heronian triangles with integer inradius n.

0, 0, 1, 1, 0, 4, 0, 2, 2, 2, 0, 6, 0, 1, 4, 3, 0, 8, 0, 6, 7, 2, 0, 17, 1, 0, 2, 8, 0, 14, 0, 3, 6, 1, 4, 17, 0, 0, 4, 12, 0, 27, 0, 4, 13, 1, 0, 27, 1, 4, 2, 4, 0, 13, 5, 14, 2, 0, 0, 32, 0, 0, 14, 4, 3, 18, 0, 5, 3, 15, 0, 41, 0, 0, 10, 4, 7, 16, 0, 18, 3, 0, 0, 60, 2, 0, 2, 18, 0, 39, 9

Offset: 1

Frank M Jackson, Aug 11 2025

nonn

If a Heronian triangle has an inradius n, and sides (x, y, z), where x <= y <= z, then the triangle is acute iff n < (x+y-z)/2.
The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5). Also, it has been proved that other than n = 3, all acute Heronian triangles have no prime inradii. For n = 3, the Heronian triangle has sides (10, 10, 12).
Empirically, it appears that the remaining occurrences of zero counts (other than 1 and the primes excluding 3) are inradii of the form 2p where p is in the set 13, 19, 29 and all other primes > 29.
The number of right integer triangles with inradius n is given by A078644, the number of obtuse Heronian triangles with inradius n is given by A386981 and the total number of Heronian triangles with inradius n is given by A120062.

			a(6) = 4, and the 4 acute Heronian triangles with inradius 6 have sides (15, 34, 35), (17, 25, 28), (17, 25, 26), (20, 20, 24).

Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.
Frank M Jackson, Mathematica program

Cf. A078644, A120062, A386981.

Mathematica
```
(* See link above. *)
```

A386981 Number of obtuse Heronian triangles with integer inradius n.

Original entry on oeis.org

0, 3, 9, 14, 12, 35, 21, 39, 44, 44, 23, 124, 28, 73, 97, 81, 30, 166, 31, 130, 169, 95, 39, 283, 59, 90, 131, 208, 33, 347, 43, 160, 196, 109, 160, 466, 35, 117, 197, 304, 41, 515, 57, 267, 354, 127, 61, 550, 110, 214, 219, 258, 44, 425, 215, 484, 265, 128, 51, 977, 41, 138, 582, 269, 169, 603, 48, 325, 252, 564, 47, 1058, 65, 133, 445, 341

Offset: 1

Frank M Jackson, Aug 11 2025

nonn

If a Heronian triangle has an inradius n, and sides (x, y, z), where x <= y <= z, then the triangle is obtuse iff n > (x+y-z)/2.
The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5).
The number of right integer triangles with inradius n is given by A078644, the number of acute Heronian triangles with inradius n is given by A386980 and the total number of Heronian triangles with inradius n is given by A120062.

			a(2) = 3, and the 3 obtuse Heronian triangles with inradius 2 have sides (6, 25, 29), (7, 15, 20), (9, 10, 17).

Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.
Frank M Jackson, Mathematica program

Cf. A078644, A120062, A386980

Mathematica
```
(* See link above. *)
```

A120569 Number of isosceles triangles with integer sides and inradius n.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 3, 0, 0, 2, 1, 0, 1, 0, 2, 2, 0, 0, 5, 0, 0, 1, 1, 0, 3, 0, 1, 1, 0, 1, 4, 0, 0, 1, 3, 0, 3, 0, 1, 2, 0, 0, 5, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 8, 0, 0, 3, 1, 0, 1, 0, 1, 1, 2, 0, 6, 0, 0, 2, 1, 0, 2, 0, 3, 1, 0, 0, 6, 0, 0, 1, 1, 0, 4, 0, 1, 1, 0, 0, 5, 0, 0, 2, 2, 0, 1, 0, 1, 5

Offset: 1

David W. Wilson, Jun 17 2006

nonn

			a(24) = 5 because 5 integer-sided isosceles triangles, namely (a,b,c) = (80,80,96), (80,85,85), (90,90,144), (130,130,240), (175,175,336), have inradius 24.

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.

David W. Wilson, Table of n, a(n) for n = 1..10000

See A120062 for sequences related to integer-sided triangles with integer inradius n.

A209432 Area A of the triangles such that A and the sides are integers and there exists at least one square inscribed in the triangle whose sides are also integers.

Original entry on oeis.org

24, 96, 216, 294, 300, 324, 384, 600, 810, 864, 1176, 1200, 1296, 1452, 1536, 1920, 1944, 2400, 2520, 2646, 2700, 2904, 2916, 3240, 3456, 4056, 4320, 4704, 4800, 4950, 5184, 5400, 5808, 6144, 6300, 6936, 7260, 7290, 7350, 7500, 7680, 7776, 8064, 8100, 8214

Offset: 1

Michel Lagneau, Mar 09 2012

nonn

Every triangle has three inscribed squares (squares in its interior such that all four of the square's vertices lie on sides of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). However, in the case of a right triangle, two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has sides of length x and the triangle has a side of length a, part of which side coincides with a side of the square, then x, a, and the triangle's area A are related according to x = 2Aa/(a^2+2A).
Property of this sequence: the numbers of the form 24*k^2 are in the sequence.
Theorem: Consider a triangle whose area A and sides (a, b, c) are integers such that there exists at least one square inscribed in this triangle whose sides x are also integers. Then, if the smallest side a = min {a, b, c} of this triangle is of the form a = 4k, k integer, then x = 3k and A = 24k^2.
Proof: Let k be an integer, and let the sides of a triangle be a = 4k, b = 13k, c = 15k. Then s = (a+b+c)/2 = 16k and A = sqrt(s(s-a)(s-b)(s-c)) = 24k^2. With x = 2Aa/(a^2+2A), we find x = 3k.

			294 is in the sequence because for (a, b, c) = (21, 28, 35) => x1 = 2*21*294/(2*294+21^2) = 12348/1029 = 12 is the integer value of the side of the square inscribed in the triangle (21, 28, 35) whose area equals 294 and whose side coincides with the side [21] of this triangle. But we also have a second square with the side x2 = 2*28*294/(2*294+28^2) = 16464/1372 = 12 whose side coincides with the side [28] of the same triangle.

Herbert Bailey and Duane Detemple, Squares inscribed in angles and triangles, Mathematics Magazine 71(4), 1998, 278-284.
Eric W. Weisstein, MathWorld: Triangle Square Inscribing

Cf. A120062, A188158, A208984.

with(numtheory):T:=array(1..1500):k:=0:nn:=500: 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): if x>0 then s:=sqrt(x) :if s=floor(s) and (irem(2*a*s,2*s+a^2) = 0 or irem(2*b*s,2*s+b^2) = 0 or irem(2*c*s,2*s+c^2) = 0) then k:=k+1:T[k]:= s: else fi:fi:od:od:od: L := [seq(T[i],i=1..k)]:L1:=convert(T,set):A:=sort(L1, `<`): print(A):

nn=500;lst={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a) (s-b) (s-c);If[0

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

Sides of the three squares: x1 = 2*A*a/(a^2+2*A); x2 = 2*A*b/(b^2+2*A); x3 = 2*A*c/(c^2+2*A).

A119672 Number of prime factors (with multiplicity) of n^4 + 3*n^2 + 1 (A057721).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 3, 3, 3, 1, 2, 3, 1, 2, 2, 1, 2, 2, 2, 2, 3, 2, 1, 3, 1, 1, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 1, 4, 2, 1, 2, 2, 2, 2, 1, 3, 2, 4, 2, 1, 3, 3, 2, 1, 2, 2, 2, 3, 1, 1, 3, 1, 2, 3, 4, 3, 1, 3, 1, 1, 2, 1, 2, 2, 2, 3, 3

Offset: 0

Jonathan Vos Post, Jun 11 2006

Cf. A001222, A057721, A120062.

Table[PrimeOmega[n^4+3n^2+1],{n,0,100}]  (* Harvey P. Dale, Apr 25 2011 *)

a(n) = Bigomega(n^4 + 3*n^2 + 1) = A001222(A057721(n)).

More terms from Harvey P. Dale, Apr 25 2011

A338393 Smallest perimeter of integer-sided triangles for which there exist exactly n triangles that have an integer inradius.

Original entry on oeis.org

12, 36, 60, 162, 108, 180, 228, 84, 132, 168, 210, 640, 252, 448, 504, 612, 462, 480, 396, 1050, 1008, 630, 672, 1632, 756, 792, 1380, 420, 1740, 1232, 1584, 1560, 1188, 1540, 2052, 1428, 1820, 840, 1620, 1320, 1890, 3612, 2912, 2280, 1092, 924, 2340, 2730, 3220

Offset: 1

Bernard Schott, Oct 28 2020

nonn

			a(1) = 12 because (3,4,5) is the smallest integer-sided triangle with an integer inradius and this integer radius = 1.
a(2) = 36 and the 2 corresponding triangles are (9,10,17) with r=2 and (9,12,15) with r=3.
a(3) = 60 and the 3 corresponding triangles are (6,25,29) with r=2, (10,24,26) with r=4 and (15,20,25) with r=5.

Eric Weisstein's World of Mathematics, Incircle.

Cf. A005044, A070201, A120062.

More terms from Amiram Eldar, Oct 28 2020

cp's OEIS Frontend

A386980 Number of acute Heronian triangles with integer inradius n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A386981 Number of obtuse Heronian triangles with integer inradius n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A120569 Number of isosceles triangles with integer sides and inradius n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A209432 Area A of the triangles such that A and the sides are integers and there exists at least one square inscribed in the triangle whose sides are also integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A119672 Number of prime factors (with multiplicity) of n^4 + 3*n^2 + 1 (A057721).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A338393 Smallest perimeter of integer-sided triangles for which there exist exactly n triangles that have an integer inradius.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions