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A357274 List of primitive triples for integer-sided triangles with angles A < B < C and C = 2*Pi/3 = 120 degrees.

%I A357274 #31 Dec 04 2022 11:58:29
%S A357274 3,5,7,7,8,13,5,16,19,11,24,31,7,33,37,13,35,43,16,39,49,9,56,61,32,
%T A357274 45,67,17,63,73,40,51,79,11,85,91,19,80,91,55,57,97,40,77,103,24,95,
%U A357274 109,13,120,127,23,120,133,65,88,133,69,91,139,56,115,151,25,143,157,75,112,163,15,161,169,104,105,181
%N A357274 List of primitive triples for integer-sided triangles with angles A < B < C and C = 2*Pi/3 = 120 degrees.
%C A357274 The only triangles with integer sides that have an angle equal to a whole number of degrees are triangles which have an angle of 60° (A335893), or an angle of 90° (A263728) or an angle of 120° as here (see Keith Selkirk link, p. 251).
%C A357274 The triples are displayed in nondecreasing order of largest side c, and if largest sides coincide then by increasing order of the smallest side a, hence, each triple (a, b, c) is in increasing order.
%C A357274 The corresponding metric relation between sides is c^2 = a^2 + a*b + b^2.
%C A357274 The triples (a, b, c) can be generated with integers u, v such that gcd(u,v) = 1 and 0 < v < u:
%C A357274 -> a = u^2 - v^2
%C A357274 -> b = 2*u*v + v^2
%C A357274 -> c = u^2 + u*v + v^2.
%C A357274 Note that side c cannot be even when the triple is primitive as here.
%C A357274 The (3, 5, 7) triangle is the only primitive triangle with a 120-degree angle and with its integer sides in arithmetic progression (A336750). This smallest triple is obtained for u = 2 and v = 1.
%C A357274 The Fermat point of these triangles is vertex C, then distance FA+FB+FC = CA+CB = b+a is an integer.
%C A357274 If (a,b,c) is a primitive 120-triple, then both (a,a+b,c) and (a+b,b,c) are 60-triples in A335893, see Emrys Read link, lemma 2 p. 302.
%H A357274 G. Julia, <a href="https://gjmaths.pagesperso-orange.fr/contenu/trcvenon.pdf">Triangles dont un angle mesure 120 degrés</a>, Problème Capes (in French).
%H A357274 Emrys Read, <a href="https://doi.org/10.1017/S0025557200179793">On integer-sided triangles containing angles of 120° or 60°</a>, Mathematical Gazette 90, July 2006, 299-305.
%H A357274 Keith Selkirk, <a href="https://doi.org/10.2307/3617258">Integer-Sided Triangles with an Angle of 120°</a>, Mathematical Gazette, Vol. 67, No. 442 (Dec., 1983), pp. 251-255.
%H A357274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LawofCosines.html">Law of Cosines</a>.
%H A357274 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_triple#Triangles_with_an_angle_of_120°">Triangles with an angle of 120°</a>, Eisenstein triple.
%e A357274 Table of triples begins:
%e A357274    3,  5,  7;
%e A357274    7,  8, 13;
%e A357274    5, 16, 19;
%e A357274   11, 24, 31;
%e A357274    7, 33, 37;
%e A357274 ............
%e A357274 (7, 8, 13) is a triple for this sequence because from the law of cosines (see link), cos(C) = (7^2 + 8^2 - 13^2)/(2*7*8) = -1/2.
%p A357274 for c from 5 to 181 by 2 do
%p A357274 for a from 3 to c-2 do
%p A357274 b := (-a + sqrt(4*c^2-3*a^2))/2;
%p A357274 if b=floor(b) and gcd(a,b)=1 and a<b then print(a,b,c); end if;
%p A357274 end do;
%p A357274 end do;
%Y A357274 Cf. A357275, A357276, A357277, A357278.
%Y A357274 Cf. also A263728, A336750, A335893 (similar with an angle of Pi/3).
%K A357274 nonn,tabf
%O A357274 1,1
%A A357274 _Bernard Schott_, Sep 22 2022
%E A357274 a(31..33) = 40,51,79 inserted by _Georg Fischer_, Dec 04 2022