Displaying 1-10 of 13 results found.
Irregular triangle read by rows: row n (n>=1) lists the distinct areas of integer-sided triangles whose area equals n times their perimeter.
+10
10
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
COMMENTS
Since the rows are long, more than the usual number of terms is shown. However, all rows are finite.
EXAMPLE
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)
...
MATHEMATICA
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 *)
CROSSREFS
For the initial term in each row see A289155, for last term see A289156.
EXTENSIONS
Title modified and inconsistent double occurrence of 168 (a(14)) deleted by Hugo Pfoertner, Mar 04 2020
Areas of integer-sided triangles whose area equals twice their perimeter.
+10
7
84, 96, 108, 120, 132, 144, 156, 168, 180, 240, 264, 300, 324, 396, 420, 684, 1224
COMMENTS
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
EXAMPLE
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,...
MATHEMATICA
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 *)
CROSSREFS
2nd row of the irregular triangle in A290451.
Areas of integer-sided triangles whose area equals 3 times their perimeter.
+10
7
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
COMMENTS
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.
EXAMPLE
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, ...
MATHEMATICA
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 *)
Areas of integer-sided triangles whose area equals 4 times their perimeter.
+10
6
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
COMMENTS
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).
EXAMPLE
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,...
MATHEMATICA
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 *)
Areas of integer-sided triangles whose area equals 5 times their perimeter.
+10
5
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
COMMENTS
There are no further terms.
EXAMPLE
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,...
Areas of integer-sided triangles whose area equals 7 times their perimeter.
+10
5
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
EXAMPLE
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,...
CROSSREFS
Cf. A332927 (listing distinct triangles with identical areas separately).
Areas of integer-sided triangles whose area equals 6 times their perimeter.
+10
4
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
CROSSREFS
Cf. A332926 (listing distinct triangles with identical areas separately).
Middle side lengths of equable Heronian triangles.
+10
2
COMMENTS
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].
CROSSREFS
Cf. A098030 (areas/perimeters), this sequence (middle side lengths), A335015 (smallest side lengths), A335016 (largest side lengths).
Smallest side lengths of equable Heronian triangles (with multiplicity).
+10
2
COMMENTS
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].
CROSSREFS
Cf. A098030 (areas/perimeters), A335013 (middle side lengths), this sequence (smallest side lengths), A335016 (largest side lengths).
Largest side lengths of equable Heronian triangles.
+10
2
COMMENTS
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], [5,12,13], [9,10,17], [7,15,20], [6,25,29].
CROSSREFS
Cf. A098030 (areas/perimeters), A335013 (middle side lengths), A335015 (smallest side lengths), this sequence (largest side lengths).
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