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A335897 Perimeters of primitive integer-sided triangles whose angles A < B < C are in arithmetic order.

%I A335897 #11 Jul 17 2020 11:12:57
%S A335897 18,20,35,36,45,56,77,90,84,110,104,126,120,143,135,182,176,189,170,
%T A335897 216,210,221,198,272,209,270,264,266,260,297,252,323,273,380,299,396,
%U A335897 351,374,368,390,378,437,350,468,425,462,360,506,494,495,432,575,476,585,464,630
%N A335897 Perimeters of primitive integer-sided triangles whose angles A < B < C are in arithmetic order.
%C A335897 The triples of sides (a,b,c) of A335893 with a < b < c are in nondecreasing order of middle side, and if middle sides coincide, then by increasing order of the largest side, and when largest sides coincide, then by increasing order of the smallest side. This sequence lists the sums a+b+c (see last example).
%C A335897 Equivalently, perimeters of primitive non-equilateral triangles that have an angle of Pi/3.
%C A335897 Also, sum a+b+c of the solutions of the Diophantine equation b^2 = a^2 - b*c + c^2 with gcd(a,b) = 1 and a < b.
%C A335897 For the corresponding primitive triples and miscellaneous properties and references, see A335893.
%C A335897 This sequence is not increasing. For example, a(8) = 90 for triangle with middle side = 31 while a(9) = 84 for triangle with middle side = 37.
%D A335897 V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.
%F A335897 a(n) = A335893(n, 1) + A335893(n, 2) + A335893(n, 3).
%F A335897 a(n) = A335894(n) + A335895(n) + A335896(n).
%e A335897 For b = 7 and c = 8, the two corresponding triangles satisfy:
%e A335897    7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8) and perimeter = 18,
%e A335897    7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8) and perimeter = 20.
%e A335897 For b = 91, there exist four corresponding triangles, two for b = 91 and c = 96 and two for b = 91 and c = 99; the four corresponding perimeters are ordered 198, 272, 209, 270 in the data because:
%e A335897    91^2 = 11^2 -11*96 +96^2, with triple (11, 91, 96) and perimeter 11+91+96 = 198,
%e A335897    91^2 = 85^2 -85*96 +96^2, with triple (85, 91, 96) and perimeter 85+91+96 = 272,
%e A335897    91^2 = 19^2 -19*99 +99^2, with triple (19, 91, 99) and perimeter 19+91+99 = 209,
%e A335897    91^2 = 80^2 -80*99 +99^2, with triple (80, 91, 99) and perimeter 80+91+99 = 270.
%p A335897 for b from 3 to 250 by 2 do
%p A335897 for c from b+1 to 6*b/5 do
%p A335897 a := (c - sqrt(4*b^2-3*c^2))/2;
%p A335897 if gcd(a,b)=1 and issqr(4*b^2-3*c^2) then print(a+b+c,2*c-a+b); end if;
%p A335897 end do;
%p A335897 end do;
%o A335897 (PARI) lista(nn) = {forstep(b=1, nn, 2, for(c=b+1, 6*b\5, if (issquare(d=4*b^2 - 3*c^2), my(a = (c - sqrtint(d))/2); if ((denominator(a)==1) && (gcd(a, b) == 1), print1(a+b+c, ", ", 2*c-a+b, ", "); ); ); ); ); } \\ _Michel Marcus_, Jul 17 2020
%Y A335897 Cf. A335893 (triples), A335894 (smallest side), A335895 (middle side), A335896 (largest side).
%K A335897 nonn
%O A335897 1,1
%A A335897 _Bernard Schott_, Jul 17 2020