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A336751 Smallest side of integer-sided triangles whose sides a < b < c are in arithmetic progression.

%I A336751 #19 Feb 28 2024 15:15:14
%S A336751 2,3,3,4,4,5,4,5,6,5,6,7,5,6,7,8,6,7,8,9,6,7,8,9,10,7,8,9,10,11,7,8,9,
%T A336751 10,11,12,8,9,10,11,12,13,8,9,10,11,12,13,14,9,10,11,12,13,14,15,9,10,
%U A336751 11,12,13,14,15,16,10,11,12,13,14,15,16,17,10,11,12,13,14
%N A336751 Smallest side of integer-sided triangles whose sides a < b < c are in arithmetic progression.
%C A336751 The triples of sides (a,b,c) with a < b < c are in increasing order of perimeter = 3*b, and if perimeter coincide, then by increasing order of the smallest side. This sequence lists the a's.
%C A336751 Equivalently: smallest side of integer-sided triangles such that b = (a+c)/2 with a < c.
%C A336751 a >= 2 and each side a appears a-1 times but not consecutively.
%C A336751 For each a = 3*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 A336751 This sequence is not increasing a(6) = 5 for triangle with perimeter = 18 and a(7) = 4 for triangle with perimeter = 21. The smallest side is not an increasing function of the perimeter of these triangles.
%C A336751 For the corresponding triples and miscellaneous properties and references, see A336750.
%D A336751 V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-290 p. 121, André Desvigne.
%H A336751 Paolo Xausa, <a href="/A336751/b336751.txt">Table of n, a(n) for n = 1..10000</a>
%F A336751 a(n) = A336750(n, 1).
%e A336751 a = 2 for only the smallest triangle (2, 3, 4).
%e A336751 a = 3 for Pythagorean triple (3, 4, 5) and also for the second triangle (3, 5, 7).
%p A336751 for b from 3 to 30 do
%p A336751 for a from b-floor((b-1)/2) to b-1 do
%p A336751 c := 2*b - a;
%p A336751 print(a);
%p A336751 end do;
%p A336751 end do;
%t A336751 Flatten[Array[Range[#-Floor[(#-1)/2], #-1] &, 20, 3]] (* _Paolo Xausa_, Feb 28 2024 *)
%Y A336751 Cf. A336750 (triples), this sequence (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter).
%Y A336751 Cf. A335894 (smallest side when triangles angles are in arithmetic progression).
%K A336751 nonn
%O A336751 1,1
%A A336751 _Bernard Schott_, Aug 15 2020