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 10 results.

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 *)

A070093 Number of acute integer triangles with perimeter n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 4, 3, 5, 4, 5, 5, 5, 6, 6, 6, 7, 7, 9, 8, 10, 9, 10, 10, 11, 12, 12, 12, 14, 13, 16, 14, 17, 16, 17, 18, 18, 20, 20, 20, 22, 22, 24, 23, 25, 26, 26, 27, 28, 30, 30, 29, 32, 31, 35, 33, 36, 36, 38, 39, 40, 40
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

An integer triangle [A070080(k) <= A070081(k) <= A070082(k)] is acute iff A070085(k) > 0.

Examples

			For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4] and [3,3,3]; two of them are acute, as 2^2+3^2<16=4^2, therefore a(9)=2.

Links

Crossrefs

Cf. A070094, A070095, A070118.
Cf. A005044, A024155, A070101.

Programs

  • Mathematica
    Table[Sum[Sum[(1 - Sign[Floor[(n - i - k)^2/(i^2 + k^2)]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 12 2019 *)

Formula

a(n) = A005044(n) - A070101(n) - A024155(n);
a(n) = A042154(n) + A070098(n).
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1-sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))). - Wesley Ivan Hurt, May 12 2019

A070122 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an acute scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

33, 45, 53, 60, 70, 83, 90, 92, 106, 114, 119, 132, 134, 142, 148, 162, 165, 168, 175, 181, 183, 197, 200, 203, 204, 218, 224, 237, 240, 245, 247, 261, 264, 267, 268, 282, 290, 293, 296, 309, 316, 317, 319, 333, 341, 345, 348
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			70 is a term because [A070080(70), A070081(70), A070082(70)]=[5<7<8], A070084(70)=gcd(5,7,8)=1, A070085(70)=5^2+7^2-8^2=25+49-64=10>0.

Links

Crossrefs

Cf. A070096, A070118, A070110, A070112.

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 && GCD[a, b, c] == 1 && a^2 + b^2 - c^2 > 0] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

A070123 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an acute scalene integer triangle with prime side lengths.

Original entry on oeis.org

240, 544, 799, 911, 1262, 1568, 2621, 2681, 2856, 3369, 3648, 4246, 5194, 5541, 6576, 6626, 6725, 7441, 7503, 7565, 7902, 7944, 8882, 8956, 9332, 9452, 9472, 9888, 9988, 10421, 10498, 10502, 11075, 11079, 11622
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			240 is a term because: [A070080(240), A070081(240), A070082(240)]=[7<11<13], A070085(240)=7^2+11^2-13^2=49+121-169=1>0.

Links

Crossrefs

Cf. A070097, A070118, A070111, A070112.

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]] &];
Position[triangles, {a_, b_, c_} /; a < b < c && AllTrue[{a, b, c}, PrimeQ] && a^2 + b^2 - c^2 > 0] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

A070125 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 6, 7, 11, 12, 15, 16, 19, 22, 23, 27, 28, 35, 39, 40, 43, 46, 47, 51, 55, 58, 63, 64, 65, 72, 73, 81, 88, 94, 95, 98, 103, 107, 108, 109, 121, 124, 135, 136, 140, 150, 151, 159, 166, 167, 170, 178, 185, 186, 189, 194, 201, 205
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(14)=22: [A070080(22), A070081(22), A070082(22)]=[3<5=5], A070084(22)=gcd(3,5,5)=1, A070085(22)=3^2+5^2-5^2=9>0.

Links

Crossrefs

Cf. A070099, A070115, A070110, A070118.

A070126 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute isosceles integer triangle with prime side lengths.

Original entry on oeis.org

3, 6, 9, 16, 22, 34, 35, 43, 46, 63, 84, 109, 124, 159, 170, 189, 201, 234, 286, 297, 350, 352, 382, 410, 450, 478, 479, 515, 527, 597, 629, 688, 708, 811, 817, 868, 900, 981, 1021, 1033, 1105, 1153, 1284, 1386, 1419, 1425
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(5)=22: [A070080(22), A070081(22), A070082(22)]=[3<5=5], A070085(22)=3^2+5^2-5^2=9>0.

Links

Crossrefs

Cf. A070100, A070115, A070111, A070118.

A070119 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an acute integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 6, 7, 11, 12, 15, 16, 19, 22, 23, 27, 28, 33, 35, 39, 40, 43, 45, 46, 47, 51, 53, 55, 58, 60, 63, 64, 65, 70, 72, 73, 81, 83, 88, 90, 92, 94, 95, 98, 103, 106, 107, 108, 109, 114, 119, 121, 124, 132, 134, 135, 136, 140, 142, 148
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(15)=33: [A070080(33), A070081(33), A070082(33)]=[4,5,6], A070084(33)=gcd(4,5,6)=1, A070085(33)=4^2+5^2-6^2=16+25-36=5>0.

Links

Crossrefs

Cf. A070094, A070122, A070125, A070118, A070110.

A070120 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle with prime side lengths.

Original entry on oeis.org

3, 6, 9, 16, 22, 34, 35, 43, 46, 63, 84, 109, 124, 159, 170, 189, 201, 234, 240, 286, 297, 350, 352, 382, 410, 450, 478, 479, 515, 527, 544, 597, 629, 688, 708, 799, 811, 817, 868, 900, 911, 981, 1021, 1033, 1105, 1153, 1262
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(6)=34: [A070080(34), A070081(34), A070082(34)]=[5,5,5], A070085(34)=5^2+5^2-5^2=25>0.

Links

Crossrefs

Cf. A070095, A070126, A070123, A070111, A070118.

A070124 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute isosceles 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, 34, 35, 38, 39, 40, 43, 46, 47, 48, 51, 54, 55, 58, 63, 64, 65, 68, 71, 72, 73, 76, 81, 84, 85, 88, 93, 94, 95, 98, 103, 107, 108, 109, 112, 117, 120, 121, 124, 129
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(13)=18: [A070080(18), A070081(18), A070082(18)]=[4=4=4], A070085(18)=4^2+4^2-4^2=16>0.

Links

Crossrefs

Cf. A070098, A070126, A070125, A070115, A070118.

A070121 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute scalene integer triangle.

Original entry on oeis.org

33, 45, 53, 60, 70, 83, 90, 92, 106, 114, 119, 132, 134, 142, 148, 162, 165, 168, 175, 181, 183, 197, 200, 203, 204, 218, 221, 224, 237, 240, 245, 247, 261, 264, 267, 268, 282, 290, 293, 296, 309, 312, 316, 317, 319, 333, 341
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(4)=60: [A070080(60), A070081(60), A070082(60)]=[4<7<8], A070085(60)=4^2+7^2-8^2=16+49-64=1>0.

Links

Crossrefs

Cf. A024154, A070122, A070123, A070112, A070118.
Showing 1-10 of 10 results.

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