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%I #18 Sep 27 2020 14:51:59
%S 0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,3,0,0,2,0,0,3,0,0,2,0,0,5,0,
%T 0,2,0,0,6,0,0,3,0,0,4,0,0,4,0,0,8,0,0,3,0,0,4,0,0,4,0,0,6,0,0,5,0,0,
%U 11,0,0,4
%N Number of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression with a perimeter = n.
%C Equivalently: number of primitive integer-sided triangles such that b = (a+c)/2 with a < c and perimeter = n.
%C As the perimeter of these triangles = 3*b where b is the middle side, a(n) >= 1 iff n = 3*b, with b >= 3.
%C When b is prime, all the triangles of perimeter n = 3*b are primitive, hence in this case: a(n) = A024164(n).
%C For the corresponding triples (primitive or not), miscellaneous properties and references, see A336750.
%F For n = 3*b, b >= 3, a(n) = A023022(b) = A000010(b)/2, otherwise a(n) = 0.
%e a(9) = 1 for the smallest such triangle (2, 3, 4).
%e a(12) = 1 for the Pythagorean triple (3, 4, 5).
%e a(15) = 2 for the two triples (3, 5, 7) and (4, 5, 6).
%e a(18) = 1 for the triple (5, 6, 7); the other triple (4, 6, 8) corresponding to a perimeter = 18 is not a primitive triple.
%Y Cf. A336750 (triples, primitive or not), A336755 (primitive triples), A336756 (perimeters of primitive triangles).
%Y Cf. A024164 (number of such triangles, primitive or not).
%Y Similar sequences: A005044 (integer-sided triangles), A024155 (right triangles), A070201 (with integral inradius).
%Y Cf. A000010, A023022.
%K nonn
%O 1,15
%A _Bernard Schott_, Sep 20 2020
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