%I #19 Feb 28 2024 15:15:14

%S 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 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 11,12,13,14,15,16,10,11,12,13,14,15,16,17,10,11,12,13,14

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

%C 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 Equivalently: smallest side of integer-sided triangles such that b = (a+c)/2 with a < c.

%C a >= 2 and each side a appears a-1 times but not consecutively.

%C 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 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 For the corresponding triples and miscellaneous properties and references, see A336750.

%D V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-290 p. 121, André Desvigne.

%H Paolo Xausa, <a href="/A336751/b336751.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A336750(n, 1).

%e a = 2 for only the smallest triangle (2, 3, 4).

%e a = 3 for Pythagorean triple (3, 4, 5) and also for the second triangle (3, 5, 7).

%p for b from 3 to 30 do

%p for a from b-floor((b-1)/2) to b-1 do

%p c := 2*b - a;

%p print(a);

%p end do;

%p end do;

%t Flatten[Array[Range[#-Floor[(#-1)/2], #-1] &, 20, 3]] (* _Paolo Xausa_, Feb 28 2024 *)

%Y Cf. A336750 (triples), this sequence (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter).

%Y Cf. A335894 (smallest side when triangles angles are in arithmetic progression).

%K nonn

%O 1,1

%A _Bernard Schott_, Aug 15 2020