A188158
Area A of the triangles such that A and the sides are integers.
Original entry on oeis.org
6, 12, 24, 30, 36, 42, 48, 54, 60, 66, 72, 84, 90, 96, 108, 114, 120, 126, 132, 144, 150, 156, 168, 180, 192, 198, 204, 210, 216, 234, 240, 252, 264, 270, 288, 294, 300, 306, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 432, 456, 462, 468, 480, 486, 504, 510, 522, 528
Offset: 1
a(3) = 24 because the area of the triangle whose sides are 4, 15, 13 is given by sqrt(p(p-4)(p-15)(p-13)) = 24, where p = (4 + 15 + 13)/2 = 16.
Cf.
A007237,
A009112,
A024153,
A024365,
A051516,
A051584,
A051585,
A055592,
A055593,
A055594,
A055595.
-
# storage of areas in T(i)
T:=array(1..4000):nn:=100:k:=1:for a from 1
to nn do: for b from 1 to nn do: for c from 1 to nn do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x1:=abs(x):s:=sqrt(x1) :else fi:if s=floor(s) then T[k]:=s:k:=k+1:else
fi:od:od:od:
# sort of T(i)
for jj from 1 to k-1 do: ii:=jj:for k1 from ii+1 to k-1 do:if T[ii]>T[k1] then ii:=k1:else fi:od: m:=T[jj]:T[jj]:=T[ii]:T[ii]:=m:od:liste:=convert(T,set):print(liste):
# second program:
isA188158 := proc(A::integer)
local Asqr, s,a,b,c ;
Asqr := A^2 ;
for s in numtheory[divisors](Asqr) do
if s^2> A then
for a from 1 to s-1 do
if modp(Asqr,s-a) = 0 then
for b from a to s-1 do
c := 2*s-a-b ;
if s*(s-a)*(s-b)*(s-c) = Asqr then
return true ;
end if;
end do:
end if;
end do:
end if;
end do:
false ;
end proc:
for n from 3 to 600 do
if isA188158(n) then
printf("%d,\n",n) ;
end if;
end do: # R. J. Mathar, May 02 2018
-
nn = 528; 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]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst] (* T. D. Noe, Mar 23 2011 *)
A120062
Number of triangles with integer sides a <= b <= c having integer inradius n.
Original entry on oeis.org
1, 5, 13, 18, 15, 45, 24, 45, 51, 52, 26, 139, 31, 80, 110, 89, 33, 184, 34, 145, 185, 103, 42, 312, 65, 96, 140, 225, 36, 379, 46, 169, 211, 116, 173, 498, 38, 123, 210, 328, 44, 560, 60, 280, 382, 134, 64, 592, 116, 228, 230, 271, 47, 452, 229, 510, 276, 134, 54
Offset: 1
a(1)=1: {3,4,5} is the only triangle with integer sides and inradius 1.
a(2)=5: {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17} are the only triangles with integer sides and inradius 2.
a(4)=A120252(1)+A120252(2)+A120252(4)=1+4+13 because 1, 2 and 4 are the factors of 4. The 1 primitive triangle with inradius n=1 is (3,4,5). The 4 primitive triangles with n=2 are (5,12,13), (9,10,17), (7,15,20), (6,25,29). The 13 primitive triangles with n=4 are (13,14,15), (15,15,24), (11,25,30), (15,26,37), (10,35,39), (9,40,41), (33,34,65), (25,51,74), (9,75,78), (11,90,97), (21,85,104), (19,153,170), (18,289,305). (Primitive means GCD(a, b, c, n)=1.)
Cf.
A078644 [Pythagorean triangles with inradius n],
A057721 [n^4+3*n^2+1].
Let S(n) be the set of triangles with integer sides a<=b<=c and inradius n. Then:
A120062(n) gives number of triangles in S(n).
A120261(n) gives number of triangles in S(n) with gcd(a, b, c) = 1.
A120252(n) gives number of triangles in S(n) with gcd(a, b, c, n) = 1.
A005408(n) = 2n+1 gives shortest short side a of triangles in S(n).
A120064(n) gives shortest middle side b of triangles in S(n).
A120063(n) gives shortest long side c of triangles in S(n).
A120570(n) gives shortest perimeter of triangles in S(n).
A120572(n) gives smallest area of triangles in S(n).
A058331(n) = 2n^2+1 gives longest short side a of triangles in S(n).
A082044(n) = n^4+2n^2+1 gives longest middle side b of triangles in S(n).
A057721(n) = n^4+3n^2+1 gives longest long side c of triangles in S(n).
A120571(n) = 2n^4+6n^2+4 gives longest perimeter of triangles in S(n).
A120573(n) = gives largest area of triangles in S(n).
Cf.
A120252 [primitive triangles with integer inradius],
A120063 [minimum of longest sides],
A057721 [maximum of longest sides],
A120064 [minimum of middle sides],
A082044 [maximum of middle sides],
A005408 [minimum of shortest sides],
A058331 [maximum of shortest sides],
A007237 [number of triangles with integer sides and area = n times perimeter].
A289218
Areas of integer-sided triangles whose area equals twice their perimeter.
Original entry on oeis.org
84, 96, 108, 120, 132, 144, 156, 168, 180, 240, 264, 300, 324, 396, 420, 684, 1224
Offset: 1
The areas 84,96,108,120,132, ... pertain respectively to triangles with sides (13,14,15), (12,16,20), (15,15,24), (10,24,26), (11,25,30), ..., equal twice their perimeter 42,48,54,60,66,...
2nd row of the irregular triangle in
A290451.
-
f[a_, b_, c_] := Block[{P = Total[{a, b, c}]/2}, Sqrt[P (P - a) (P - b) (P - c)]]; Sort@ Map[f @@ # &, Select[Union@ Map[Sort, Tuples[Range@ 200, {3}]], f @@ # == 4 Total@ # &] ] (* Michael De Vlieger, Jul 03 2017 *)
A289155
Smallest area of triangle with integer sides and area = n times perimeter.
Original entry on oeis.org
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
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.
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
A120063
Shortest side c of all integer-sided triangles with sides a<=b<=c and inradius n.
Original entry on oeis.org
5, 10, 12, 15, 25, 24, 35, 30, 36, 39, 55, 45, 65, 63, 53, 60, 85, 68, 95, 75, 77, 88, 115, 85, 125, 130, 108, 105, 145, 106, 155, 120, 132, 170, 137, 135, 185, 190, 156, 150, 205, 154, 215, 165, 159, 230, 235, 170, 245, 195, 204, 195, 265, 204, 200, 195, 228, 290
Offset: 1
a(1)=5 because the only triangle with integer sides and inradius 1 is {3,4,5}; its longest side is 5.
a(2)=10: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their longest sides is min(13,10,29,20,17)=10.
- 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.
See
A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf.
A120062 [triangles with integer inradius],
A120252 [primitive triangles with integer inradius],
A057721 [maximum of longest sides],
A058331 [maximum of shortest sides],
A120064 [minimum of middle sides],
A082044 [maximum of middle sides],
A005408 [minimum of shortest sides],
A007237.
A120064
Shortest side b of all integer-sided triangles with sides a<=b<=c and inradius n.
Original entry on oeis.org
4, 8, 10, 14, 20, 20, 28, 28, 30, 39, 44, 40, 52, 56, 50, 56, 68, 60, 76, 70, 70, 87, 92, 80, 100, 100, 90, 97, 116, 100, 124, 112, 110, 136, 120, 120, 148, 152, 130, 140, 164, 140, 172, 154, 150, 184, 188, 160, 196, 174, 170, 182, 212, 180, 196, 189, 190, 232, 236
Offset: 1
a(1)=2 because the only triangle with integer sides a<=b<c and inradius 1 is {3,4,5}; its middle side is 4.
a(2)=8: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their middle sides is min(12,8,25,15,10)=8.
- 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.
Cf.
A120062 [triangles with integer inradius],
A120252 [primitive triangles with integer inradius],
A057721 [maximum of longest sides],
A120063 [minimum of longest sides],
A058331 [maximum of shortest sides],
A082044 [maximum of middle sides],
A005408 [minimum of shortest sides],
A007237.
See
A120062 for sequences related to integer-sided triangles with integer inradius n.
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
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.
-
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
A371973
a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.
Original entry on oeis.org
1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 13, 19, 14, 21, 19, 23, 20, 27, 23, 30, 27, 32, 29, 35, 32, 39, 34, 44, 39, 48, 43, 52, 47, 55, 51, 60, 53, 63, 59, 69, 58, 74, 67, 78, 73, 84, 75, 90, 81, 92, 88, 101, 91, 108, 93, 112, 106
Offset: 3
See the formula section for the relationships with
A026810,
A070083,
A135622 (which has many crossrefs related to areas of triangles).
-
A2(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
a371973(n) = {my (A=List()); forpart (v=n, listput(A, A2(v[1],v[2],v[3])), [1,(n-1)\2], [3,3]); #Set(A)};
-
def A371973(n): return len(set((2*(b+c)-n)*(n-2*b)*(n-2*c) for c in range((n+2)//3, (n+1)//2) for b in range((n-c+1)//2, c+1))) # David Radcliffe, Aug 01 2025
A332689
Number of distinct areas of integer-sided triangles whose area equals n times their perimeter.
Original entry on oeis.org
5, 17, 41, 41, 47, 127, 77, 81, 171, 132, 99, 283, 94, 205, 349, 158, 115, 457, 122, 296, 530, 267, 134, 546, 219, 260, 428, 471, 130, 953, 144, 264, 613, 332, 557, 1031, 139, 346, 614, 600, 162, 1381, 169, 562, 1132, 348, 186, 1000, 363, 593, 688, 571, 164, 1123
Offset: 1
For n = 2, there are 18 different (noncongruent) Heronian triangles whose area equals twice their perimeter, so A007237(2) = 18. However, two of those 18 triangles share the area 168. So there are only 17 distinct areas. Therefore, a(2) = 17.
-
a[k_] := Block[{v={},r,s,t}, Do[ If[r <= s && 4 k^2 < r s <= 12 k^2 && IntegerQ[t = 4 k^2 (r + s)/(r s - 4 k^2)] && t >= s, AppendTo[v, r + s + t]], {r, Floor[2 Sqrt[3] k]}, {s, Floor[4 k^2/r], Ceiling[12 k^2/r]}]; Length@ Union@ v]; Array[a, 20] (* Giovanni Resta, Mar 04 2020 *)
-
from math import sqrt
def A332689(n):
L = []; k = 4*n*n
for x in range(1, int(2*sqrt(3)*n) + 1):
for y in range(max(int(k/x) + 1, x), int((k + 2*n*sqrt(k + x*x))/x) + 1):
if k*(x+y)%(x*y-k) == 0:
s = x + y + k*(x+y)//(x*y-k)
if s not in L: L.append(s)
return len(L) # Ya-Ping Lu, Dec 28 2023
A332926
2*a(n) are the perimeters of distinct triangles with integer sides i <= j <= k, whose area equals 6 times their perimeter. Terms occurring more than once belong to different triangles.
Original entry on oeis.org
63, 64, 65, 68, 70, 70, 72, 77, 77, 80, 81, 82, 84, 85, 88, 90, 91, 93, 95, 99, 105, 108, 110, 112, 112, 115, 117, 125, 126, 126, 128, 135, 136, 140, 143, 145, 152, 152, 153, 154, 160, 165, 168, 174, 180, 182, 182, 187, 198, 203, 203, 203, 205, 205, 208, 217
Offset: 1
The terms of
A332879, divided by 12, are all terms of this sequence, but omitting distinct triangles with identical perimeters.
Showing 1-10 of 10 results.
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