A098030 -id:A098030 - cp's OEIS frontend

A290451 Irregular triangle read by rows: row n (n>=1) lists the distinct areas of integer-sided triangles whose area equals n times their perimeter.

24, 30, 36, 42, 60, 84, 96, 108, 120, 132, 144, 156, 168, 180, 240, 264, 300, 324, 396, 420, 684, 1224, 192, 204, 210, 216, 240, 252, 264, 270, 324, 330, 336, 378, 384, 408, 420, 456, 462, 480, 504, 522, 540, 546, 624, 690, 714, 780, 792, 840, 876, 966, 990, 1176, 1248, 1320, 1380, 1806, 2394, 2460, 3120, 4446, 8436, 336, 360, 384, 432, 456, 480, 528, 576, 624, 672, 720, 840, 960, 1056, 1176

Offset: 1

N. J. A. Sloane, Aug 06 2017

Since the rows are long, more than the usual number of terms is shown. However, all rows are finite.

			The first few rows of the triangle are:
(n=1) 24, 30, 36, 42, 60
(n=2) 84, 96, 108, 120, 132, 144, 156, 168, 180, 240, 264, 300, 324, 396, 420, 684, 1224
(n=3) 192, 204, 210, 216, 240, 252, 264, 270, 324, 330, 336, 378, ... (truncated)
(n=4) 336, 360, 384, 432, 456, 480, 528, 576, 624, 672, 720, 840, ... (truncated)
(n=5) 540, 600, 630, 660, 750, 810, 840, 900, 930, 1050, 1080, ... (truncated)
(n=6) 756, 768, 780, 816, 840, 864, 924, 960, 972, 984, 1008, ... (truncated)
(n=7) 1134, 1176, 1344, 1386, 1470, 1596, 1680, 1764, 1848, 1890, ... (truncated)
...

For the initial term in each row see A289155, for last term see A289156.
Rows are: n=1: A098030, n=2: A289218, n=3: A289219, n=4: A289220, n=5: A289221, n=6: A332879, n=7: A289253.
Cf. A332689 (row lengths).

row[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]}]; 2 k Union@ v]; Join @@ Array[row, 4] (* Giovanni Resta, Mar 04 2020 *)

Title modified and inconsistent double occurrence of 168 (a(14)) deleted by Hugo Pfoertner, Mar 04 2020

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

Zhining Yang, Jun 28 2017

There are no further terms.

One term, 168, corresponds to exactly two different triangles, namely [14, 30, 40] and [10, 35, 39], both with perimeter 84. The remaining terms correspond to unique triangles. - Jeppe Stig Nielsen, Mar 04 2020

			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,...

Cf. A098030, A007237.
2nd row of the irregular triangle in A290451.
Cf. A332922.

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 *)

Duplicate term 168 (previous a(9)) removed by Jeppe Stig Nielsen, Mar 04 2020

A289219 Areas of integer-sided triangles whose area equals 3 times their perimeter.

Original entry on oeis.org

192, 204, 210, 216, 240, 252, 264, 270, 324, 330, 336, 378, 384, 408, 420, 456, 462, 480, 504, 522, 540, 546, 624, 690, 714, 780, 792, 840, 876, 966, 990, 1176, 1248, 1320, 1380, 1806, 2394, 2460, 3120, 4446, 8436

Offset: 1

Zhining Yang, Jun 28 2017

There are no further terms.
For a(3)=210, there are 2 solutions (20,21,29),(17,25,28);
For a(11)=336, there are 2 solutions (14,48,50),(24,35,53);
For a(16)=456, a(22)=546, there are 2 solutions respectively too.

			The areas 192,204,210,216,240, ... pertain respectively to triangles with sides (20,20,24), (17,25,26), (20,21,29), (18,24,30), (16,30,34), ..., equal 3 times their perimeter 64,68,70,72,80, ...

Cf. A098030, A289218.

A row of the 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@ 150, {3}]], f @@ # == 3 Total@# &] ] (* Michael De Vlieger, Jul 03 2017 *)

A289220 Areas of integer-sided triangles whose area equals 4 times their perimeter.

Original entry on oeis.org

336, 360, 384, 432, 456, 480, 528, 576, 624, 672, 720, 840, 960, 1056, 1176, 1200, 1224, 1296, 1584, 1680, 1944, 2064, 2088, 2184, 2328, 2520, 2736, 2856, 3240, 3696, 4440, 4488, 4896, 5256, 6600, 7728, 9240, 9360, 9384, 17688, 34320

Offset: 1

Zhining Yang, Jun 28 2017

There are no further terms.
For a(10)=672, there are 2 solutions: (28,60,80), (20,70,78).
For a(12)=840, there are 3 solutions: (35,73,102), (25,84,101), (21,89,100).

			The areas 336,360,384,432,456, ... pertain respectively to triangles with sides (26,28,30), (25,29,36), (24,32,40), (30,30,48), (25,38,51), ..., equal 4 times their perimeter 84,90,96,108,114,...

Cf. A098030, A289218, A289219.

A row of the 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 *)

A289221 Areas of integer-sided triangles whose area equals 5 times their perimeter.

Original entry on oeis.org

540, 600, 630, 660, 750, 810, 840, 900, 930, 1050, 1080, 1320, 1380, 1500, 1560, 1590, 1740, 2040, 2070, 2280, 2310, 2520, 2580, 2970, 3150, 3240, 3720, 4020, 4350, 4530, 4620, 5460, 6270, 6300, 7260, 7560, 7800, 7980, 11730, 12210, 14040, 18870, 22260, 27030, 27300, 52530, 103020

Offset: 1

Zhining Yang, Jun 28 2017

There are no further terms.

			The areas 540,600,630,660,750, ... pertain respectively to triangles with sides (30,39,39), (30,40,50), (28,45,53), (26,51,55), (25,60,65)...., equal 5 times their perimeter 108,120,126,132,150,...

Cf. A098030, A289218, A289219, A289220.

A row of the triangle in A290451.

A289253 Areas of integer-sided triangles whose area equals 7 times their perimeter.

Original entry on oeis.org

1134, 1176, 1344, 1386, 1470, 1596, 1680, 1764, 1848, 1890, 2016, 2058, 2184, 2310, 2394, 2520, 2604, 2856, 2940, 3024, 3360, 3696, 3780, 3864, 4032, 4242, 4368, 4536, 4830, 5292, 5544, 5712, 6006, 6090, 6216, 6258, 6510, 6636, 6720

Offset: 1

Zhining Yang, Jun 29 2017

			The areas 1134,1176,1344,1386,1470, ... pertain respectively to triangles with sides (39,60,63), (42,56,70), (40,68,84), (36,77,85), (35,84,91)...., equal 7 times their perimeter 162,168,192,198,210,...

Zhining Yang, Table of n, a(n) for n = 1..77

Cf. A098030, A289218, A289219, A289220, A289221.
A row of the triangle in A290451.
Cf. A332927 (listing distinct triangles with identical areas separately).

A374594 Areas of trapezoids with integer sides and height whose area equals their perimeter.

Original entry on oeis.org

16, 18, 18, 20, 20, 24, 30, 30, 36, 48, 70, 90, 180, 180, 420, 528, 870, 1170, 2610

Offset: 1

Felix Huber, Jul 13 2024

A trapezoid is a quadrilateral with at least one pair of parallel sides.

Conjecture: in this sequence are only four terms which belong to trapezoids with exactly one pair of parallel sides: a(2) = 18, a(4) = 20, a(6) = 24, a(7) = 30.

			See attached illustration of the terms a(1) to a(11).

Felix Huber, Illustration of terms a(1) to a(11)
Felix Huber, Sides and heights of the trapezoids belonging to the terms a(1) to a(19)
Eric Weisstein's World of Mathematics,Trapezoid.
Wikipedia, Trapezoid.

Cf. A098030, A181945, A189415, A214602, A272459, A335187, A340858, A348143, A360790.

with(NumberTheory):
A374594:=proc(k);
  local K,L,S,T,i,a,c,x,y,h,b,d;
  L:=map(x->x/2, Divisors(2*k) minus {1, 2});
  S:=[];
  T:=[];
  K:=[];
  for i to numelems(L) do
    for c to L[i] do
      a:=2*L[i]-c;
      h:=k/L[i];
      x:=0;
      while x^2<(k-a-c)^2-h^2 do
        if issqr(x^2+h^2) then
          d:=isqrt(x^2+h^2);
          b:=k-a-c-d;
          y:=a-c-x;
          if h^2+y^2=b^2 then
            S:=[a,b,c,d];
            S:=sort(S);
            if member(S,T)=false then
              T:=[op(T),S];
              K:=[op(K),k];
            fi;
          fi;
        fi;
        x:=x+1;
      od;
    od;
  od;
  if numelems(K)>0 then
    return op(K)
  fi;
end proc;
seq(A374594(k),k=1..3000);

Corrected by Felix Huber, Dec 04 2024

A332879 Areas of integer-sided triangles whose area equals 6 times their perimeter.

Original entry on oeis.org

756, 768, 780, 816, 840, 864, 924, 960, 972, 984, 1008, 1020, 1056, 1080, 1092, 1116, 1140, 1188, 1260, 1296, 1320, 1344, 1380, 1404, 1500, 1512, 1536, 1620, 1632, 1680, 1716, 1740, 1824, 1836, 1848, 1920, 1980, 2016, 2088, 2160, 2184, 2244, 2376, 2436, 2460

Offset: 1

Hugo Pfoertner, Mar 02 2020

Hugo Pfoertner, Table of n, a(n) for n = 1..127, providing the full sequence.

6th row in A290451.
Cf. A098030, A289155, A289218, A289219, A289220, A289221, A289253.
Cf. A332926 (listing distinct triangles with identical areas separately).

A335013 Middle side lengths of equable Heronian triangles.

Original entry on oeis.org

8, 10, 12, 15, 25

Offset: 1

Wesley Ivan Hurt, May 19 2020

Equable Heronian triangles are triangles with integer-sides, integer area and whose area is equal to their perimeter. There are exactly five, [6,8,10], [9,10,17], [5,12,13], [7,15,20], [6,25,29].

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A098030 (areas/perimeters), this sequence (middle side lengths), A335015 (smallest side lengths), A335016 (largest side lengths).

A335015 Smallest side lengths of equable Heronian triangles (with multiplicity).

Original entry on oeis.org

5, 6, 6, 7, 9

Offset: 1

Wesley Ivan Hurt, May 19 2020

Equable Heronian triangles are triangles with integer-sides, integer area and whose area is equal to their perimeter. There are exactly five, [5,12,13], [6,8,10], [6,25,29], [7,15,20], [9,10,17].

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A098030 (areas/perimeters), A335013 (middle side lengths), this sequence (smallest side lengths), A335016 (largest side lengths).

cp's OEIS Frontend

A290451 Irregular triangle read by rows: row n (n>=1) lists the distinct areas of integer-sided triangles whose area equals n times their perimeter.

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A289218 Areas of integer-sided triangles whose area equals twice their perimeter.

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A289219 Areas of integer-sided triangles whose area equals 3 times their perimeter.

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A289220 Areas of integer-sided triangles whose area equals 4 times their perimeter.

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A289221 Areas of integer-sided triangles whose area equals 5 times their perimeter.

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A289253 Areas of integer-sided triangles whose area equals 7 times their perimeter.

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A374594 Areas of trapezoids with integer sides and height whose area equals their perimeter.

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A332879 Areas of integer-sided triangles whose area equals 6 times their perimeter.

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A335013 Middle side lengths of equable Heronian triangles.

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A335015 Smallest side lengths of equable Heronian triangles (with multiplicity).

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