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A351802
a(n) = A351477(n) * FB where F is the Fermat point of a primitive integer-sided triangle ABC with A < B < C < 2*Pi/3 and FA + FB + FC = A336329(n).
5
264, 325, 1064, 27265, 6528, 34200, 12376, 1015, 8512, 11520, 8415, 1656, 116025, 8415, 17575, 56448, 81928, 6765, 107712, 106128, 43953, 60903, 235008, 311885, 3105, 32571, 571648, 411320, 9499, 4991, 1800, 13875, 1894144, 16320, 402375, 42735, 805, 218925
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 FB = 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 FB = a(n).
FB is the middle length with FC < FB < FA (remember a < b < c).
FORMULA
a(n) = A351476(n) - A351801(n) - A351803(n).
FB = sqrt(((2*a*c)^2 - (a^2+c^2-d^2)^2)/3) / d. - Jinyuan Wang, Feb 19 2022
EXAMPLE
For the 2nd triple in A336328, i.e., (73, 88, 95), we get A336329(2) = FA + FB + FC = 440/7 + 325/7 + 264/7 = 147, hence A351477(2) = 7 and a(2) = 325.
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*c)^2 - (a^2+c^2-d^2)^2)/3)/d), ", "); ); ); ); ); } \\ Michel Marcus, Mar 01 2022
CROSSREFS
Cf. A336328 (primitive triples), A336329 (FA + FB + FC), A336330 (smallest side), A336331 (middle side), A336332 (largest side), A336333 (perimeter), A351801 (FA numerator), this sequence (FB numerator), A351803 (FC numerator), A351477 (common denominator of FA, FB, FC), A351476 (other 'FA + FB + FC').
Sequence in context: A253694 A253701 A255804 * A050240 A105683 A160971
KEYWORD
nonn
AUTHOR
Bernard Schott, Feb 19 2022
EXTENSIONS
More terms from Jinyuan Wang, Feb 19 2022
STATUS
approved