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

1,1

Comments

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

Links

Table of n, a(n) for n=1..78.

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.

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).

Sequence in context: A075422 A230195 A098030 * A068544 A284174 A292982

Adjacent sequences: A290448 A290449 A290450 * A290452 A290453 A290454

Keyword

nonn,tabf

Author

N. J. A. Sloane, Aug 06 2017

Extensions

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

Status

approved