A289156 -id:A289156 - 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

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

Hugo Pfoertner, Mar 02 2020

There are A007237(6) = 139 integer-sided triangles with 6*area = perimeter, with terms 70, 77, 112, 126, 152, 182, 205, 260, 513, and 798 corresponding to pairs, and terms 203 to triples of such triangles with different sides.

Hugo Pfoertner, Table of n, a(n) for n = 1..139

The terms of A332879, divided by 12, are all terms of this sequence, but omitting distinct triangles with identical perimeters.

Cf. A007237, A289155, A289156.

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions