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 73 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).

A070085 a(n) = A070080(n)^2 + A070081(n)^2 - A070082(n)^2.

Original entry on oeis.org

1, 1, 4, 1, -1, 4, 1, -3, 9, 4, 2, 1, -5, -7, 9, 4, 0, 16, 1, -7, -11, 9, 7, 4, -2, -4, 16, 1, -9, -15, 9, -17, 5, 25, 4, -4, -8, 16, 14, 1, -11, -19, 9, -23, 3, 1, 25, 4, -6, -12, 16, -14, 12, 36, 1, -13, -23, 9, -29, 1, -31, -3, 25, 23, 4, -8
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

The integer triangle [A070080(n)<=A070081(n)<=A070082(n)] is acute iff a(n)>0, right iff a(n)=0 and obtuse iff a(0)<0.

Links

Crossrefs

Cf. A070093, A070101, A024155, A070118, A070127, A070136.

Programs

  • Mathematica
    maxPer = m = 22;
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]]&];
#[[1]]^2 + #[[2]]^2 - #[[3]]^2& /@ triangles (* Jean-François Alcover, Jul 31 2018 *)

A070084 Greatest common divisor of sides of integer triangles [A070080(n), A070081(n), A070082(n)], sorted by perimeter, sides lexicographically ordered.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

a(n)>1 iff there exists a smaller similar triangle [A070080(k), A070081(k), A070082(k)] with kA070080(n)=A070080(k)*a(n), A070081(n)=A070081(k)*a(n) and A070082(n)=A070082(k)*a(n).

Links

Crossrefs

Cf. A051493, A005044, A070091, A070094, A070102, A070109, A070110, A070113, A070116, A070119, A070128, A070137.

Programs

  • Mathematica
    maxPer = 22; maxSide = Floor[(maxPer - 1)/2]; order[{a_, b_, c_}] := (a + b + c)*maxPer^3 + a*maxPer^2 + b*maxPer + c; triangles = Reap[Do[If[a + b + c <= maxPer && 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]]; GCD @@@ Sort[triangles, order[#1] < order[#2] &] (* Jean-François Alcover, May 27 2013 *)

Formula

a(n) = GCD(A070080(n), A070081(n), A070082(n)).

A070086 Areas of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 3, 4, 4, 4, 2, 4, 4, 6, 5, 6, 7, 3, 5, 5, 7, 8, 6, 7, 8, 9, 3, 6, 6, 9, 7, 10, 11, 7, 9, 10, 11, 12, 4, 6, 8, 10, 8, 12, 12, 14, 8, 10, 12, 13, 12, 15, 16, 4, 7, 9, 12, 10, 14, 10, 15, 16, 17, 9, 12, 13, 15, 14, 17, 18, 19, 5, 8, 10
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

Triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))] have integer areas = a(A070142(k)) = A070149(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*(s-3)*(s-5)*(s-6)) = sqrt(7*(7-3)*(7-5)*(7-6)) = sqrt(7*4*2*1) = sqrt(56) = 7.48331, rounded = 7.

Links

Crossrefs

Cf. A051516, A055595, A046131.
The sides are given by A070080, A070081, A070082.
See A135622 for values signifying the precise area and further crossrefs.

Programs

  • Mathematica
    m = 50; (* 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)] // Round];
area /@ triangles (* Jean-François Alcover, Oct 03 2021 *)

Formula

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

A070110 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 27, 28, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 72, 73, 74, 75, 77
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

A070084(a(k)) = gcd(A070080(a(k)), A070081(a(k)), A070082(a(k))) = 1;
all integer triangles [A070080(a(k)), A070081(a(k)), A070082(a(k))] are mutually nonisomorphic.

Examples

			13 is a term: [A070080(13), A070081(13), A070082(13)]=[2,4,5], A070084(13)=gcd(2,4,5)=1.

Links

Crossrefs

Cf. A051493, A070113, A070116, A070119, A070122, A070125, A070128, A070131, A070134, A070137, A070084.

Programs

  • Mathematica
    m = 50 (* 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]] &];
Position[triangles, {a_, b_, c_} /; GCD[a, b, c] == 1] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

A070118 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 15, 16, 18, 19, 22, 23, 24, 27, 28, 31, 33, 34, 35, 38, 39, 40, 43, 45, 46, 47, 48, 51, 53, 54, 55, 58, 60, 63, 64, 65, 68, 70, 71, 72, 73, 76, 81, 83, 84, 85, 88, 90, 92, 93, 94, 95, 98, 103, 106, 107, 108
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(21)=33: [A070080(33), A070081(33), A070082(33)]=[4,5,6], A070085(33)=4^2+5^2-6^2=16+25-36=5>0.

Links

Crossrefs

Cf. A070093, A070119, A070120, A070121, A070122, A070123, A070124, A070125, A070126.

Programs

  • Mathematica
    m = 50; (* 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]] &];
Position[triangles, {a_, b_, c_} /; a^2 + b^2 - c^2 > 0] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

A070127 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse integer triangle.

Original entry on oeis.org

5, 8, 13, 14, 20, 21, 25, 26, 29, 30, 32, 36, 37, 41, 42, 44, 49, 50, 52, 56, 57, 59, 61, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 82, 86, 87, 89, 91, 96, 97, 99, 100, 101, 102, 104, 105, 110, 111, 113, 115, 118, 122, 123, 125, 126, 127
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(10)=30: [A070080(30), A070081(30), A070082(30)]=[3,5,7], A070085(30)=3^2+5^2-7^2=9+25-49=-15<0.

Links

Crossrefs

Cf. A070101, A070129, A070130, A070131, A070132, A070133, A070134, A070135, A070128, A070085.

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]]&];
Position[triangles, {a_, b_, c_} /; a^2 + b^2 - c^2 < 0] // Flatten (* Jean-François Alcover, Oct 11 2021 *)

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 *)

A070112 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle.

Original entry on oeis.org

8, 13, 17, 20, 21, 25, 29, 30, 33, 36, 37, 41, 42, 44, 45, 49, 50, 53, 56, 57, 59, 60, 62, 66, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 86, 87, 89, 90, 92, 96, 97, 99, 100, 101, 102, 105, 106, 110, 111, 113, 114, 115, 116, 119, 122
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(17)=50: [A070080(50), A070081(50), A070082(50)]=[4<6<8].

Links

Crossrefs

Cf. A005044, A070113, A070114, A070121, A070122, A070123, A070130, A070131, A070132.

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]] &];
Position[triangles, {a_, b_, c_} /; a < b < c] // Flatten (* Jean-François Alcover, Oct 12 2021 *)
Showing 1-10 of 73 results. Next

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