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.
Original entry on oeis.org
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
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.
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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
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
Cf.
A332926 (listing distinct triangles with identical areas separately).
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.
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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 *)
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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
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 1-4 of 4 results.
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