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 1-10 of 13 results. Next

A070080 Smallest side of integer triangles [a(n) <= A070081(n) <= A070082(n)], sorted by perimeter, lexicographically ordered.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 3, 2, 3, 1, 2, 3, 3, 2, 3, 4, 1, 2, 3, 3, 4, 2, 3, 4, 4, 1, 2, 3, 3, 4, 4, 5, 2, 3, 4, 4, 5, 1, 2, 3, 3, 4, 4, 5, 5, 2, 3, 4, 4, 5, 5, 6, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 2, 3, 4, 4, 5, 5, 6, 6, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 2, 3, 4, 4, 5, 5
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A046128, A055594, A069597.
Cf. A070081, A070082, A070083, A070084, A070085, A070086.
Cf. A316841, A316843, A316844, A316845 (sides (i,j,k) with j + k > i >= j >= k >= 1).
Cf. A331244, A331245, A331246 (similar, but triangles sorted by radius of enclosing circle), A331251, A331252, A331253 (triangles sorted by area), A331254, A331255, A331256 (triangles sorted by radius of circumcircle).

Programs

  • Mathematica
    m = 55 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2&];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
triangles[[All, 1]] (* Jean-François Alcover, Jun 12 2012, updated Jul 09 2017 *)

Formula

a(n) = A070083(n) - A070082(n) - A070081(n).

A070081 Middle side of integer triangles [A070080(n) <= a(n) <= A070082(n)], sorted by perimeter, sides lexicographically ordered.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 3, 5, 4, 3, 4, 5, 4, 4, 6, 5, 4, 5, 4, 6, 5, 4, 5, 7, 6, 5, 6, 4, 5, 5, 7, 6, 5, 6, 5, 8, 7, 6, 7, 5, 6, 5, 6, 8, 7, 6, 7, 5, 6, 6, 9, 8, 7, 8, 6, 7, 5, 6, 7, 6, 9, 8, 7, 8, 6, 7, 6, 7, 10, 9, 8, 9, 7, 8, 6, 7, 8, 6, 7, 7, 10, 9, 8, 9, 7
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A046129, A055593, A069595, A069594, A069598.
Cf. A070080, A070082, A070083, A070084, A070085, A070086.

Programs

  • Mathematica
    m = 55 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2&];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
triangles[[All, 2]] (* Jean-François Alcover, Jul 09 2017 *)

Formula

a(n) = A070083(n) - A070080(n) - A070082(n).

A070082 Largest side of integer triangles [A070080(n) <= A070081(n) <= a(n)], sorted by perimeter, sides lexicographically ordered.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 3, 4, 4, 5, 5, 5, 4, 5, 5, 4, 6, 6, 6, 5, 5, 6, 6, 6, 5, 7, 7, 7, 6, 7, 6, 5, 7, 7, 7, 6, 6, 8, 8, 8, 7, 8, 7, 7, 6, 8, 8, 8, 7, 8, 7, 6, 9, 9, 9, 8, 9, 8, 9, 8, 7, 7, 9, 9, 9, 8, 9, 8, 8, 7, 10, 10, 10, 9, 10, 9, 10, 9, 8, 9, 8, 7, 10, 10
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A046130, A055592, A069598, A069597.
Cf. A070080, A070081, A070083, A070084, A070085, A070086.

Programs

  • Mathematica
    m = 55 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2&];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
triangles[[All, 3]] (* Jean-François Alcover, Jul 09 2017 *)

Formula

a(n) = A070083(n) - A070080(n) - A070081(n).

A070142 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer area.

Original entry on oeis.org

17, 39, 52, 116, 212, 252, 269, 368, 370, 372, 375, 493, 561, 587, 659, 839, 850, 862, 957, 972, 1156, 1186, 1196, 1204, 1297, 1582, 1599, 1629, 1912, 1920, 1955, 1971, 1988, 2115, 2352, 2555, 2574, 2713, 2774, 2778, 2790
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(2)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

Links

Crossrefs

Cf. A070088, A070149, A070143, A070144, A070145, A051516, A069594, A069597, A070136, A070137, A070209.

Programs

  • Mathematica
    maxPerim = 100; maxSide = Floor[(maxPerim - 1)/2]; order[{a_, b_, c_}] := (a + b + c)*maxPerim^3 + a*maxPerim^2 + b*maxPerim + c; triangles = Reap[ Do[ If[ a + b + c <= maxPerim && c - b < a < c + b && b - a < c < b + a && c - a < b < c + a, Sow[{a, b, c}]], {a, 1, maxSide}, {b, a, maxSide}, {c, b, maxSide}]][[2, 1]]; stri = Sort[ triangles, order[#1] < order[#2]&]; area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p*(p - a)*(p - b)*(p - c)]]; Position[ stri, tri_ /; IntegerQ[area[tri]]] // Flatten (* Jean-François Alcover, Feb 22 2013 *)

A070149 Areas of integer Heronian triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))].

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			A070142(2)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore a(2)=A070086(39)=area=4*3=12.

Links

Crossrefs

Cf. A070142, A070210.

Programs

  • Mathematica
    m = 500 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
area[{a_, b_, c_}] := With[{p = (a+b+c)/2}, Sqrt[p(p-a)(p-b)(p-c)]];
Select[area /@ triangles, IntegerQ] (* Jean-François Alcover, Oct 12 2021 *)

Formula

a(n) = A070086(A070142(n)).

A070200 Inradii of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

Triangles [A070080(A070209(n)), A070081(A070209(n)), A070082(A070209(n))] have integer inradii = a(A070209(k))= A070210(k).

Examples

			[A070080(25), A070081(25), A070082(25)] = [3,5,6] and s = A070083(25)/2 = (3+5+6)/2 = 7: a(25) = sqrt((s-3)*(s-5)*(s-6)/7) = sqrt((7-3)*(7-5)*(7-6)/7) = sqrt(4*2*1/7) = sqrt(8/7) = 1.069, rounded = 1.

Links

Crossrefs

Cf. A070086.

Formula

a(n) = sqrt((s-u)*(s-v)*(s-w)/s), where u=A070080(n), v=A070081(n), w=A070082(n) and s=A070083(n)/2=(u+v+w)/2.

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).

A070145 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with integer area.

Original entry on oeis.org

39, 52, 269, 372, 375, 862, 957, 972, 1204, 1955, 1971, 1988, 2790, 2796, 3818, 5374, 6522, 6880, 6881, 6921, 7234, 7310, 7341, 7360, 9198, 9207, 10272, 14506, 15101, 15177, 15237, 21289, 21493, 21540, 21552, 21589
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(1)=39: [A070080(39), A070081(39), A070082(39)] = [5=5<6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

Links

Crossrefs

Cf. A070142, A070144, A070139.

A070144 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle with integer area.

Original entry on oeis.org

17, 116, 212, 252, 368, 370, 493, 561, 587, 659, 839, 850, 1156, 1186, 1196, 1297, 1582, 1599, 1629, 1912, 1920, 2115, 2352, 2555, 2574, 2713, 2774, 2778, 3251, 3473, 3728, 3746, 3751, 4286, 4298, 4307, 4313, 4319
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(2)=116: [A070080(116), A070081(116), A070082(116)] = [6<8<10], area^2 = s*(s-6)*(s-8)*(s-10) with s=A070083(116)/2=(6+8+10)/2=12, area^2=12*6*4*2=64*9 is an integer square, therefore A070086(116)=area=8*3=24.

Links

Crossrefs

Cf. A070142, A070145.

A070146 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle with integer area.

Original entry on oeis.org

39, 269, 375, 587, 862, 972, 1196, 1955, 1988, 2352, 2555, 2796, 3818, 4319, 4406, 5378, 6522, 6808, 6880, 6890, 6921, 7234, 7360, 8193, 9159, 9207, 10272, 14545, 15004, 15061, 15101, 15216, 15237, 15943, 16502
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(1)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6]: A070085(39)=5^2+5^2-6^2=14>0 and area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

Links

Crossrefs

Cf. A070142, A070136, A070147, A070140.
Showing 1-10 of 13 results. Next

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

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