A051493 Triangles with perimeter n and relatively prime integer side lengths.

0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 4, 2, 5, 2, 5, 4, 8, 4, 10, 6, 9, 6, 14, 8, 15, 9, 16, 12, 21, 11, 24, 16, 22, 16, 27, 18, 33, 20, 31, 24, 40, 23, 44, 30, 39, 30, 52, 32, 54, 35, 52, 42, 65, 38, 65, 48, 64, 49, 80, 48, 85, 56, 77, 64, 90, 58, 102, 72, 93, 69, 114, 72, 120, 81

Offset

1,7

Comments

From Peter Munn, Jul 26 2017: (Start)

The triangles that meet the conditions are listed by nondecreasing n in A070110.

Without the requirement for relatively prime side lengths, this sequence becomes A005044.

Counting the triangles by longest side instead of perimeter, this sequence becomes A123323.

a(n) = A070094(n) + A070102(n) + A070109(n).

(End)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Transforms

Formula

Moebius transform of A005044.

Example

There are 3 triangles with integer-length sides and perimeter 9: 1-4-4, 2-3-4, 3-3-3. 3-3-3 is omitted because isomorphic to 1-1-1, so a(9)=2.

Mathematica

nmax = 100;
A005044[n_] := Quotient[n^2 + 6n Mod[n, 2] + 24, 48];
A = Array[A005044, nmax];
mob[m_, n_] := If[ Mod[m, n] == 0, MoebiusMu[m/n], 0];
Reap[Do[Sow[Sum[mob[n, d] A[[d]], {d, 1, n}]], {n, 1, nmax}]][[2, 1]] (* Jean-François Alcover, Oct 05 2021 *)

Crossrefs

Cf. A005044, A057887, A070110, A123323.

Equivalent sequences, restricted to subsets: A070091 (isosceles), A070094 (acute), A070102 (obtuse), A070109 (right-angled), A070138 (with integer area), A070202 (with integer inradius).

Sequence in context: A160974 A187718 A029196 * A338201 A029173 A002331

Adjacent sequences: A051490 A051491 A051492 * A051494 A051495 A051496

Keyword

nonn

Author

Neil Fernandez

Extensions

Corrected and extended with formula by Christian G. Bower, Nov 15 1999

Formula updated due to change to referenced sequence, and definition clarified by Peter Munn, Jul 26 2017

Status

approved