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A336754 Perimeters in increasing order of integer-sided triangles whose sides a < b < c are in arithmetic progression.

%I A336754 #24 Feb 29 2024 14:32:54
%S A336754 9,12,15,15,18,18,21,21,21,24,24,24,27,27,27,27,30,30,30,30,33,33,33,
%T A336754 33,33,36,36,36,36,36,39,39,39,39,39,39,42,42,42,42,42,42,45,45,45,45,
%U A336754 45,45,45,48,48,48,48,48,48,48,51,51,51,51,51,51,51,51
%N A336754 Perimeters in increasing order of integer-sided triangles whose sides a < b < c are in arithmetic progression.
%C A336754 Equivalently: perimeters of integer-sided triangles such that b = (a+c)/2 with a < c.
%C A336754 As perimeter = 3 * middle side, these perimeters p are all multiple of 3, and each term p appears floor((p-3)/6) = A004526((p-3)/3) consecutively.
%C A336754 For each perimeter = 12*k with k>0, there exists one right integer triangle whose triple is (3k, 4k, 5k).
%C A336754 For the corresponding primitive triples, miscellaneous properties and references, see A336750.
%D A336754 V. Lespinard and R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-290 p. 121, André Desvigne.
%H A336754 Paolo Xausa, <a href="/A336754/b336754.txt">Table of n, a(n) for n = 1..10000</a>
%F A336754 a(n) = A336750(n, 1) + A336750(n, 2) + A336750(n, 3).
%F A336754 a(n) = 3 * A307136(n).
%e A336754 Perimeter = 9 only for the smallest triangle (2, 3, 4).
%e A336754 Perimeter = 12 only for Pythagorean triple (3, 4, 5).
%e A336754 Perimeter = 15 for the two triples (3, 5, 7) and (4, 5, 6).
%p A336754 for b from 3 to 30 do
%p A336754 for a from b-floor((b-1)/2) to b-1 do
%p A336754 c := 2*b - a;
%p A336754 print(a+b+c);
%p A336754 end do;
%p A336754 end do;
%t A336754 A336754[n_] := 3*Ceiling[2*Sqrt[n+Round[Sqrt[n]]]]; Array[A336754, 100] (* or *)
%t A336754 Flatten[Array[ConstantArray[3*#, Floor[(#-1)/2]] &, 19, 3]] (* _Paolo Xausa_, Feb 29 2024 *)
%Y A336754 Cf. A004526.
%Y A336754 Cf. A336750 (triples), A336751 (smallest side), A307136 (middle side), A336753 (largest side), this sequence (perimeter), A024164 (number of such triangles whose perimeter = n), A336755 (primitive triples).
%Y A336754 Cf. A335897 (perimeters when angles A, B and C are in arithmetic progression).
%K A336754 nonn
%O A336754 1,1
%A A336754 _Bernard Schott_, Aug 31 2020