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

%I A070091 #11 Oct 05 2021 13:30:45
%S A070091 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,
%T A070091 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,
%U A070091 12,6,17,8,10,6,18,6,18,9,10,9,14,6,20,8,13
%N A070091 Number of isosceles integer triangles with perimeter n and relatively prime side lengths.
%C A070091 a(n) = A051493(n) - A005044(n-6).
%H A070091 Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%e A070091 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.
%t A070091 m = 81 (* max perimeter *);
%t A070091 sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
%t A070091 triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &] ;
%t A070091 a[n_] := Count[triangles, t_ /; Total[t] == n && Length[Union[t]] < 3 && GCD @@ t == 1];
%t A070091 Table[a[n], {n, 1, m}] (* _Jean-François Alcover_, Oct 05 2021 *)
%Y A070091 Cf. A070080, A070081, A070082, A059169, A070099, A070107, A070084, A070116.
%K A070091 nonn
%O A070091 1,7
%A A070091 _Reinhard Zumkeller_, May 05 2002