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A336753 Largest side of integer-sided triangles whose sides a < b < c are in arithmetic progression.

%I A336753 #19 Feb 28 2024 15:14:54
%S A336753 4,5,7,6,8,7,10,9,8,11,10,9,13,12,11,10,14,13,12,11,16,15,14,13,12,17,
%T A336753 16,15,14,13,19,18,17,16,15,14,20,19,18,17,16,15,22,21,20,19,18,17,16,
%U A336753 23,22,21,20,19,18,17,25,24,23,22,21,20,19,18,26,25,24,23,22,21,20,19
%N A336753 Largest side of integer-sided triangles whose sides a < b < c are in arithmetic progression.
%C A336753 The triples of sides (a,b,c) with a < b < c are in increasing order of perimeter = 3*b, and if perimeters coincide, then by increasing order of the smallest side. This sequence lists the c's.
%C A336753 Equivalently: largest side of integer-sided triangles such that b = (a+c)/2 with a < c.
%C A336753 c >= 4 and each largest side c appears floor((c-1)/3) = A002264(c-1) times but not consecutively.
%C A336753 For each c = 5*k, k>=1, there exists exactly one right triangle (3*k, 4*k, 5*k) whose sides a < b < c are in arithmetic progression.
%C A336753 For the corresponding primitive triples and miscellaneous properties and references, see A336750.
%D A336753 V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-290 p. 121, André Desvigne.
%H A336753 Paolo Xausa, <a href="/A336753/b336753.txt">Table of n, a(n) for n = 1..10000</a>
%F A336753 a(n) = A336750(n, 3).
%e A336753 c = 4 only for the smallest triangle (2, 3, 4).
%e A336753 c = 5 only for Pythagorean triple (3, 4, 5).
%e A336753 c = 6 only for triple (4, 5, 6).
%e A336753 c = 7 for the two triples (3, 5, 7) and (5, 6, 7).
%p A336753 for b from 3 to 30 do
%p A336753 for a from b-floor((b-1)/2) to b-1 do
%p A336753 c := 2*b - a;
%p A336753 print(c);
%p A336753 end do;
%p A336753 end do;
%t A336753 Flatten[Array[2*#-Range[#-Floor[(#-1)/2], #-1] &, 20, 3]] (* _Paolo Xausa_, Feb 28 2024 *)
%Y A336753 Cf. A336750 (triples), A336751 (smallest side), A307136 (middle side), this sequence (largest side), A336754 (perimeter).
%Y A336753 Cf. A335896 (largest side when triangles angles are in arithmetic progression).
%K A336753 nonn
%O A336753 1,1
%A A336753 _Bernard Schott_, Aug 25 2020