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

Author

Neil Fernandez

Keywords

• nonn

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)

Examples

			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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Transforms

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).

Programs

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 *)

Formula

Moebius transform of A005044.

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