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

%I A051493 #18 Oct 05 2021 22:12:15
%S A051493 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,
%T A051493 24,16,22,16,27,18,33,20,31,24,40,23,44,30,39,30,52,32,54,35,52,42,65,
%U A051493 38,65,48,64,49,80,48,85,56,77,64,90,58,102,72,93,69,114,72,120,81
%N A051493 Triangles with perimeter n and relatively prime integer side lengths.
%C A051493 From _Peter Munn_, Jul 26 2017: (Start)
%C A051493 The triangles that meet the conditions are listed by nondecreasing n in A070110.
%C A051493 Without the requirement for relatively prime side lengths, this sequence becomes A005044.
%C A051493 Counting the triangles by longest side instead of perimeter, this sequence becomes A123323.
%C A051493 a(n) = A070094(n) + A070102(n) + A070109(n).
%C A051493 (End)
%H A051493 Michael De Vlieger, <a href="/A051493/b051493.txt">Table of n, a(n) for n = 1..10000</a>
%H A051493 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A051493 Moebius transform of A005044.
%e A051493 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.
%t A051493 nmax = 100;
%t A051493 A005044[n_] := Quotient[n^2 + 6n Mod[n, 2] + 24, 48];
%t A051493 A = Array[A005044, nmax];
%t A051493 mob[m_, n_] := If[ Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t A051493 Reap[Do[Sow[Sum[mob[n, d] A[[d]], {d, 1, n}]], {n, 1, nmax}]][[2, 1]] (* _Jean-François Alcover_, Oct 05 2021 *)
%Y A051493 Cf. A005044, A057887, A070110, A123323.
%Y A051493 Equivalent sequences, restricted to subsets: A070091 (isosceles), A070094 (acute), A070102 (obtuse), A070109 (right-angled), A070138 (with integer area), A070202 (with integer inradius).
%K A051493 nonn
%O A051493 1,7
%A A051493 _Neil Fernandez_
%E A051493 Corrected and extended with formula by _Christian G. Bower_, Nov 15 1999
%E A051493 Formula updated due to change to referenced sequence, and definition clarified by _Peter Munn_, Jul 26 2017