cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

N. J. A. Sloane, Aug 06 2017

Keywords

Comments

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

Examples

			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)
...

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).

Programs

  • 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 *)

Extensions

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

A289219 Areas of integer-sided triangles whose area equals 3 times their perimeter.

Original entry on oeis.org

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
Offset: 1

Views

Author

Zhining Yang, Jun 28 2017

Keywords

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.

Examples

			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, ...

Crossrefs

Cf. A098030, A289218.
A row of the triangle in A290451.

Programs

  • 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 *)

A289220 Areas of integer-sided triangles whose area equals 4 times their perimeter.

Original entry on oeis.org

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
Offset: 1

Views

Author

Zhining Yang, Jun 28 2017

Keywords

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).

Examples

			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,...

Crossrefs

Cf. A098030, A289218, A289219.
A row of the triangle in A290451.

Programs

  • 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 *)

A289221 Areas of integer-sided triangles whose area equals 5 times their perimeter.

Original entry on oeis.org

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
Offset: 1

Views

Author

Zhining Yang, Jun 28 2017

Keywords

Comments

There are no further terms.

Examples

			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,...

Crossrefs

Cf. A098030, A289218, A289219, A289220.
A row of the triangle in A290451.

A289253 Areas of integer-sided triangles whose area equals 7 times their perimeter.

Original entry on oeis.org

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
Offset: 1

Views

Author

Zhining Yang, Jun 29 2017

Keywords

Examples

			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,...

Links

Crossrefs

Cf. A098030, A289218, A289219, A289220, A289221.
A row of the triangle in A290451.
Cf. A332927 (listing distinct triangles with identical areas separately).

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

Views

Author

Hugo Pfoertner, Mar 02 2020

Keywords

Links

Crossrefs

6th row in A290451.
Cf. A098030, A289155, A289218, A289219, A289220, A289221, A289253.
Cf. A332926 (listing distinct triangles with identical areas separately).

A332922 2*a(n) are the perimeters of distinct triangles with integer sides i <= j <= k, whose area equals 2 times their perimeter. Terms occurring more than once belong to different triangles.

Original entry on oeis.org

21, 24, 27, 30, 33, 36, 39, 42, 42, 45, 60, 66, 75, 81, 99, 105, 171, 306
Offset: 1

Views

Author

Hugo Pfoertner, Mar 02 2020

Keywords

Comments

There are 18 integer-sided triangles with 2*area = perimeter, with terms 42 corresponding to a pair of such triangles with different sides.

Crossrefs

Equals A289218/2.
Cf. A289219, A289220, A289221, A332879, A289253 (all merging terms for triangles with identical perimeters, but different sides).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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