A335893 - OEIS

%I #64 Sep 21 2022 12:01:06

%S 3,7,8,5,7,8,7,13,15,8,13,15,5,19,21,16,19,21,11,31,35,24,31,35,7,37,

%T 40,33,37,40,13,43,48,35,43,48,16,49,55,39,49,55,9,61,65,56,61,65,32,

%U 67,77,45,67,77,17,73,80,63,73,80,40,79,91,51,79,91,11,91,96

%N Primitive triples for integer-sided triangles whose angles A < B < C are in arithmetic progression.

%C The triples are displayed in nondecreasing order of middle side, and if middle sides coincide then by increasing order of the largest side, hence, each triple (a, b, c) is in increasing order.

%C These three properties below are equivalent:

%C -> integer-sided triangles whose angles A < B < C are in arithmetic progression,

%C -> integer-sided triangles such that B = (A+C)/2 with A < C,

%C -> integer-sided triangles such that A < B < C with B = Pi/3.

%C When A < B < C are in arithmetic progression with B = A + phi and C = B + phi, then 0 < phi < Pi/3.

%C The corresponding metric relation between sides is b^2 = a^2 - a*c + c^2.

%C There exists such primitive triangle iff b^2 is an odd square term of A024612. Hence, the first few middle sides b are 7, 13, 19, 31, 37, 43, 49, 61, 67, ... and b is a term of A004611 \ {1}. Indeed, b cannot be even if the triple is primitive.

%C As B = Pi/3 and C runs from Pi/3 to 2*Pi/3, sin(C) gets a maximum when C = Pi/2 with sin(C) = 1, hence, from law of sines (see link): b/sin(B) = c/sin(C), and c < b/sin(Pi/3) = b * 2/sqrt(3) < 6*b/5. This bound is used in the PARI and Maple programs below.

%C When triple (a, b, c) is solution, then triple (c-a, b, c) is another solution. Hence, for each b odd solution, there exist 2 triples with same middle side b and same largest side c.

%C The common tangent to the nine-point circle and the incircle of a triangle ABC is parallel to the Euler line iff angles A < B < C are in arithmetic progression (see Crux Mathematicorum for Indian team selection). - _Bernard Schott_, Apr 14 2022

%C These triples are called (primitive) Eisenstein triples (Wikipedia). - _Bernard Schott_, Sep 21 2022

%D V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.

%H Bill Sands, <a href="https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv36n5.pdf">Indian team selection test 2007, question 5</a>, Crux Mathematicorum, Vol. 36, No. 5 (2010), p. 278.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LawofCosines.html">Law of Cosines</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LawofSines.html">Law of Sines</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_triple">Eisenstein triple</a>.

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.

%e (3, 7, 8) is a triple for this sequence because from law of cosines (see link), cos(A) = (7^2 + 8^2 - 3^2)/(2*7*8) = 13/14, cos(B) = (8^2 + 3^2 - 7^2)/(2*8*3) = 1/2 and cos(C) = (3^2 + 7^2 - 8^2)/(2*3*7) = -1/7; then, (A+C)/2 = ( arccos(13/14) + arccos(-1/7) )/2 = Pi/3 = B.

%e Also, arccos(13/14) ~ 21.787 degrees, arccos(1/2) = 60 degrees, arccos(-1/7) ~ 98.213 degrees, so B-A = C-B ~ 38.213 degrees, hence (A, B, C) are in arithmetic progression.

%e 5^2 - 5*8 + 8^2 = 7^2, hence (5, 7, 8) is another triple for triangle whose angles A < B < C are in arithmetic progression.

%p for b from 3 to 250 by 2 do

%p for c from b+1 to 6*b/5 do

%p a := (c - sqrt(4*b^2-3*c^2))/2;

%p if gcd(a,b)=1 and issqr(4*b^2-3*c^2) then print(a,b,c,c-a,b,c); end if;

%p end do;

%p end do;

%o (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), print(a, ", ", b, ", ", c, ", "); print(c-a, ", ", b, ", ", c, ", ");););););} \\ _Michel Marcus_, Jul 15 2020

%Y Cf. A004611, A024612.

%Y Cf. A335894 (smallest side), A335895 (middle side), A335896 (largest side), A335897 (perimeter).

%Y Cf. A103606 (primitive Pythagorean triples), A335034 (primitive triples for triangles with two perpendicular medians).

%K nonn,tabf

%O 1,1

%A _Bernard Schott_, Jun 29 2020