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 1-4 of 4 results.

A070102 Number of obtuse integer triangles with perimeter n and relatively prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 1, 3, 2, 3, 2, 5, 3, 6, 2, 8, 5, 9, 5, 9, 6, 11, 6, 14, 9, 14, 9, 17, 11, 19, 12, 19, 15, 23, 13, 27, 18, 26, 16, 32, 20, 33, 21, 34, 26, 40, 23, 42, 29, 42, 29, 50, 32, 53, 35, 48, 41, 58, 37, 64, 45, 60, 42, 71
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

a(n) = A051493(n) - A070094(n) - A070109(n).

Examples

			For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4] and [3,3,3]; only one of them is obtuse: 2^2+3^2<16=4^2 and GCD(2,3,4)=1, therefore a(9)=1.

Links

Crossrefs

Cf. A070080, A070081, A070082, A070101, A051493, A070104, A070107, A070084, A070128.

A070134 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

5, 14, 32, 52, 61, 104, 118, 133, 146, 163, 202, 242, 246, 266, 314, 342, 404, 437, 467, 472, 504, 542, 547, 577, 619, 625, 714, 757, 801, 807, 853, 907, 957, 1015, 1022, 1082, 1139, 1145, 1265, 1278, 1335, 1414, 1475
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(5)=61: [A070080(61), A070081(61), A070082(61)]=[5=5<9], A070084(69)=gcd(5,5,9)=1, A070085(61)=5^2+5^2-9^2=25+25-81=-31<0.

Links

Crossrefs

Cf. A070107, A070110, A070127, A070115.

A070091 Number of isosceles integer triangles with perimeter n and relatively prime side lengths.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 1, 1, 3, 1, 3, 1, 2, 2, 4, 2, 5, 2, 2, 2, 6, 2, 5, 3, 5, 3, 7, 2, 8, 4, 4, 4, 6, 3, 9, 4, 6, 4, 10, 4, 11, 5, 6, 5, 12, 4, 10, 5, 8, 6, 13, 4, 10, 6, 8, 7, 15, 4, 15, 7, 10, 8, 12, 6, 17, 8, 10, 6, 18, 6, 18, 9, 10, 9, 14, 6, 20, 8, 13
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

a(n) = A051493(n) - A005044(n-6).

Examples

			For n=15 there are A005044(15)=7 integer triangles: [1,7,7], [2,6,7], [3,5,7], [3,6,6], [4,4,7], [4,5,6] and [5,5,5]: four are isosceles: [1<7=7], [3<6=6], [4=4<7] and [5=5=5], but GCD(3,6,6)>1 and GCD(5,5,5)>1, therefore a(15)=2.

Links

Crossrefs

Cf. A070080, A070081, A070082, A059169, A070099, A070107, A070084, A070116.

Programs

  • Mathematica
    m = 81 (* 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]] &] ;
a[n_] := Count[triangles, t_ /; Total[t] == n && Length[Union[t]] < 3 && GCD @@ t == 1];
Table[a[n], {n, 1, m}] (* Jean-François Alcover, Oct 05 2021 *)

A070106 Number of integer triangles with perimeter n which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 3, 4, 4, 3, 4, 4
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

a(n)=A070101(n)-A024156(n); a(n)=A059169(n)-A070098(n).

Examples

			For n=11 there are A005044(11)=4 integer triangles: [1,5,5], [2,4,5], [3,3,5] and [3,4,4]; only one of the two obtuses ([2,4,5] and [3,3,5]) is also isosceles; therefore a(11)=1.

Links

Crossrefs

Cf. A070080, A070081, A070082, A059169, A070107, A070108, A070133.
Showing 1-4 of 4 results.

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