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A336750 Triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.

%I A336750 #46 May 06 2022 13:13:51
%S A336750 2,3,4,3,4,5,3,5,7,4,5,6,4,6,8,5,6,7,4,7,10,5,7,9,6,7,8,5,8,11,6,8,10,
%T A336750 7,8,9,5,9,13,6,9,12,7,9,11,8,9,10,6,10,14,7,10,13,8,10,12,9,10,11,6,
%U A336750 11,16,7,11,15,8,11,14,9,11,13,10,11,12,7,12,17,8,12,16
%N A336750 Triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.
%C A336750 The triples are displayed in increasing order of perimeter, and if perimeters coincide then by increasing order of the smallest side, hence, each triple (a, b, c) is in increasing order.
%C A336750 Equivalently: triples of integer-sided triangles such that b = (a+c)/2 with a < c.
%C A336750 As the perimeter of these triangles = 3*b, the triples are also displayed in increasing order of middle side.
%C A336750 When a < b < c are in arithmetic progression with b - a = c - b = x, then 1 <= x <= floor((b-1)/2), hence, there exist for each side b >= 3, floor((b-1)/2) = A004526(b) triangles whose sides a < b < c are in arithmetic progression.
%C A336750 The only right integer-sided triangles such that a < b < c are in arithmetic progression correspond to the Pythagorean triples (3k, 4k, 5k) with k > 0.
%C A336750 There do not exist triangles whose sides a < b < c and angles A < B < C are both in arithmetic progression.
%C A336750 Three geometrical properties about these triangles, even if they are not integer-sided:
%C A336750    1) tan(A/2) * tan(C/2) = 1/3,
%C A336750    2) r = h_b/3, where r is the inradius and h_b the length of the altitude through B,
%C A336750    3) The line (IG) is parallel to side (AC), where I is the incenter and G is the centroid of the triangle.
%D A336750 V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-288 p. 120, André Desvigne.
%H A336750 Michel Marcus, <a href="/A336750/b336750.txt">Table of n, a(n) for n = 1..16650</a> (Rows 1 to 5550)
%H A336750 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer triangle</a>.
%F A336750 T(n,1) = A336751(n); T(n,2) = A307136(n); T(n,3) = A336753(n).
%F A336750 A336754(n) = T(n,1) + T(n,2) + T(n,3).
%e A336750 The smallest such triangle is (2, 3, 4).
%e A336750 The only triangle with perimeter = 12 corresponds to the Pythagorean triple: (3, 4, 5).
%e A336750 There exist two triangles with perimeter = 15 corresponding to triples (3, 5, 7) and (4, 5, 6).
%e A336750 There exist also two triangles with perimeter = 18 corresponding to triples (4, 6, 8) and (5, 6, 7).
%e A336750 The table begins:
%e A336750   2, 3, 4;
%e A336750   3, 4, 5;
%e A336750   3, 5, 7;
%e A336750   4, 5, 6;
%e A336750   4, 6, 8;
%e A336750   5, 6, 7;
%e A336750   4, 7, 10;
%e A336750   5, 7, 9;
%e A336750   6, 7, 8;
%p A336750 for b from 3 to 20 do
%p A336750 for a from b-floor((b-1)/2) to b-1 do
%p A336750 c := 2*b - a;
%p A336750 print(a,b,c);
%p A336750 end do;
%p A336750 end do;
%t A336750 Block[{nn = 12, a, b, c}, Reap[Do[Do[Sow@ {a, b, 2 b - a}, {a, b - Floor[(b - 1)/2], b - 1}], {b, 3, nn}]][[-1, 1]] ] // Flatten (* _Michael De Vlieger_, Oct 15 2020 *)
%o A336750 (PARI) tabf(nn) = {for (b = 3, nn, for (a = b-floor((b-1)/2), b-1, my(c = 2*b - a); print(a, " ", b, " ", c);););} \\ _Michel Marcus_, Sep 08 2020
%Y A336750 Cf. A336751 (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter), A024164 (number of triangles with perimeter = n), A336755 (primitive triples), A336756 (perimeter of primitive triangles), A336757 (number of primitive triangles with perimeter = n).
%Y A336750 Cf. A004526 (number of triangles with middle side = b).
%Y A336750 Cf. A103605 (similar, with Pythagorean triples).
%Y A336750 Cf. A335893 (similar, with A, B, C in arithmetic progression).
%K A336750 nonn,tabf
%O A336750 1,1
%A A336750 _Bernard Schott_, Aug 03 2020