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 1-6 of 6 results.

A055595 Area of triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

6, 12, 12, 24, 48, 30, 60, 54, 24, 84, 48, 36, 60, 120, 108, 66, 42, 96, 84, 126, 60, 108, 192, 90, 150, 84, 168, 120, 36, 204, 240, 210, 210, 60, 120, 216, 132, 300, 96, 336, 72, 192, 144, 240, 480, 294, 84, 252, 360, 432, 114, 156, 180, 210, 420, 120, 210, 420
Offset: 1

Views

Author

Henry Bottomley, May 26 2000

Keywords

Comments

This is the ordering of triangles used for A316841.

Links

Crossrefs

The sides are given by A055592, A055593, A055594.
Cf. A046131, A070786, A316841, A316853.
Range of values: A188158.

Programs

  • Mathematica
    max = 42; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055595 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 4]](* Jean-François Alcover, Jun 12 2012 *)

Formula

a(n) = sqrt(s(n)*(s(n)-A055592(n))*(s(n)-A055593(n))*(s(n)-A055594(n))) where s(n) = (A055592(n)+A055593(n)+A055594(n))/2 i.e. half the perimeter of the triangle

A070787 Number of triangles with sides whose squares are integers and with positive integer area and longest side of length sqrt(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 0, 2, 1, 4, 0, 1, 3, 0, 2, 5, 5, 2, 0, 13, 0, 0, 0, 2, 9, 8, 1, 1, 9, 4, 0, 10, 0, 10, 2, 12, 11, 0, 3, 23, 14, 0, 0, 1, 13, 0, 0, 5, 5, 18, 5, 32, 18, 2, 2, 2, 0, 19, 0, 13, 16, 0, 1, 20, 35, 0, 0, 42, 0, 4, 0, 23, 24, 23, 9, 1, 0, 8, 0, 44, 10, 27, 0, 1, 48, 0, 9, 2, 27, 25, 3
Offset: 1

Views

Author

Henry Bottomley, May 07 2002

Keywords

Examples

			a(13)=3 since the 3 triangles with sides {sqrt(13), sqrt(5), sqrt(4)}, {sqrt(13), sqrt(8), sqrt(1)} and {sqrt(13), sqrt(9), sqrt(4)} have areas 2, 1 and 3 respectively.

Links

Crossrefs

Cf. A054875, A070783, A070784, A070785, A070786.

A070783 Square of longest side of triangles with sides whose squares are integers and with positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

4, 5, 5, 8, 8, 9, 10, 10, 10, 10, 12, 13, 13, 13, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 24, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 26, 26, 26, 27, 28, 29, 29, 29, 29, 29, 29, 29, 29, 29
Offset: 1

Views

Author

Henry Bottomley, May 07 2002

Keywords

Examples

			a(6)=9 since the triangle with sides sqrt(9), sqrt(8) and sqrt(5) has area 3.

Links

Crossrefs

Cf. A055592, A070784, A070785, A070786, A070787.

A070784 Square of second longest side of triangles with sides whose squares are integers and with positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

2, 4, 5, 4, 5, 8, 4, 8, 10, 10, 6, 5, 8, 9, 12, 15, 5, 8, 10, 13, 13, 5, 8, 13, 16, 17, 10, 16, 8, 9, 10, 10, 13, 13, 16, 17, 17, 17, 18, 20, 20, 12, 15, 8, 13, 13, 16, 17, 17, 20, 20, 25, 10, 10, 16, 18, 20, 20, 20, 26, 24, 14, 9, 13, 16, 20, 20, 20, 25, 25, 29, 12, 24, 30, 30, 10
Offset: 1

Views

Author

Henry Bottomley, May 07 2002

Keywords

Examples

			a(6)=8 since the triangle with sides sqrt(9), sqrt(8) and sqrt(5) has area 3.

Links

Crossrefs

Cf. A055593, A070783, A070785, A070786, A070787.

A070785 Square of shortest side of triangles with sides whose squares are integers and with positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

2, 1, 4, 4, 1, 5, 2, 2, 4, 8, 6, 4, 1, 4, 3, 12, 5, 8, 2, 5, 13, 4, 5, 8, 1, 4, 4, 10, 4, 5, 2, 10, 1, 5, 4, 1, 9, 13, 2, 8, 16, 12, 3, 5, 4, 8, 9, 4, 16, 1, 5, 20, 4, 8, 2, 8, 2, 10, 18, 4, 15, 14, 8, 4, 5, 1, 13, 17, 4, 8, 16, 6, 6, 12, 24, 10, 5, 16, 9, 4, 1, 17, 10, 26, 5, 8, 10, 4, 2, 10, 4
Offset: 1

Views

Author

Henry Bottomley, May 07 2002

Keywords

Examples

			a(6)=5 since the triangle with sides sqrt(9), sqrt(8) and sqrt(5) has area 3.

Links

Crossrefs

Cf. A055594, A070783, A070784, A070786, A070787.

A135622 16*Area^2 of integer triangles [A070080(n),A070081(n),A070082(n)].

Original entry on oeis.org

3, 15, 48, 35, 63, 128, 63, 135, 243, 240, 320, 99, 231, 275, 495, 384, 576, 768, 143, 351, 455, 819, 975, 560, 896, 1008, 1344, 195, 495, 675, 1215, 735, 1575, 1875, 768, 1280, 1536, 2048, 2304, 255, 663, 935, 1683, 1071, 2295, 2499, 2975, 1008, 1728
Offset: 1

Views

Author

Franz Vrabec, Feb 29 2008

Keywords

Examples

			A070080(4)=1, A070081(4)=3, A070082(4)=3, so a(4)=(1+3+3)*(-1+3+3)*(1-3+3)*(1+3-3)=35.

Crossrefs

See the formula section for the relationships with A070080, A070081, A070082, A070086.
Cf. A317182 (range of values), A331011 (nonunique values), A331250 (counts triangles by area).
Cf. A316853 (with terms ordered as for A316841), and using this order for other sets of triangles: A046131, A055595, A070786.

Formula

a(n)=(u+v+w)*(-u+v+w)*(u-v+w)*(u+v-w), where u=A070080(n), v=A070081(n), w=A070082(n).
A070086(n) = round(sqrt(a(n))/4).
Showing 1-6 of 6 results.

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