OFFSET
1,1
COMMENTS
Inspired by Project Euler, Problem 143 (see link) where such a triangle is called a "Torricelli triangle".
For the corresponding primitive triples, miscellaneous properties and references, see A336328.
Equivalently, a(n) is the numerator of the fraction FC = a(n) / A351477(n).
Also, if F is the Fermat point of a triangle ABC with A < B < C < 2*Pi/3, where AB, BC, CA, FA, FB and FC are all positive integers, then, when FA + FB + FC = d = A351476(n), we have FC = a(n).
FC is the smallest length with FC < FB < FA (remember a < b < c).
LINKS
FORMULA
FC = sqrt(((2*a*b)^2 - (a^2 + b^2 - d^2)^2)/3) / d. - Jinyuan Wang, Feb 19 2022
EXAMPLE
PROG
(PARI) lista(nn) = {my(d); for(c=4, nn, for(b=ceil(c/sqrt(3)), c-1, for(a=1+(sqrt(4*c^2-3*b^2)-b)\2, b-1, if(gcd([a, b, c])==1 && issquare(d=6*(a^2*b^2+b^2*c^2+c^2*a^2)-3*(a^4+b^4+c^4)) && issquare(d=(a^2+b^2+c^2+sqrtint(d))/2), d = sqrtint(d); print1(numerator(sqrtint(((2*a*b)^2 - (a^2 + b^2 - d^2)^2)/3)/d), ", "); ); ); ); ); } \\ Michel Marcus, Mar 02 2022
CROSSREFS
Cf. A336328 (primitive triples), A336329 (FA + FB + FC), A336330 (smallest side), A336331 (middle side), A336332 (largest side), A336333 (perimeter), A351801 (FA numerator), A351802 (FB numerator), this sequence (FC numerator), A351477 (common denominator of FA, FB, FC), A351476 (FA + FB + FC other).
KEYWORD
nonn
AUTHOR
Bernard Schott, Feb 19 2022
EXTENSIONS
More terms from Jinyuan Wang, Feb 19 2022
STATUS
approved