A290451 - OEIS

%I #22 Mar 07 2020 01:49:58

%S 24,30,36,42,60,84,96,108,120,132,144,156,168,180,240,264,300,324,396,

%T 420,684,1224,192,204,210,216,240,252,264,270,324,330,336,378,384,408,

%U 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

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

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

%e The first few rows of the triangle are:

%e (n=1) 24, 30, 36, 42, 60

%e (n=2) 84, 96, 108, 120, 132, 144, 156, 168, 180, 240, 264, 300, 324, 396, 420, 684, 1224

%e (n=3) 192, 204, 210, 216, 240, 252, 264, 270, 324, 330, 336, 378, ... (truncated)

%e (n=4) 336, 360, 384, 432, 456, 480, 528, 576, 624, 672, 720, 840, ... (truncated)

%e (n=5) 540, 600, 630, 660, 750, 810, 840, 900, 930, 1050, 1080, ... (truncated)

%e (n=6) 756, 768, 780, 816, 840, 864, 924, 960, 972, 984, 1008, ... (truncated)

%e (n=7) 1134, 1176, 1344, 1386, 1470, 1596, 1680, 1764, 1848, 1890, ... (truncated)

%e ...

%t 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 *)

%Y For the initial term in each row see A289155, for last term see A289156.

%Y Rows are: n=1: A098030, n=2: A289218, n=3: A289219, n=4: A289220, n=5: A289221, n=6: A332879, n=7: A289253.

%Y Cf. A332689 (row lengths).

%K nonn,tabf

%O 1,1

%A _N. J. A. Sloane_, Aug 06 2017

%E Title modified and inconsistent double occurrence of 168 (a(14)) deleted by _Hugo Pfoertner_, Mar 04 2020