A229838 Consider all primitive 60-degree triangles with sides A < B < C. The sequence gives the values of A.

3, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 104, 105

Offset: 1

Author

Colin Barker, Oct 01 2013

Keywords

• nonn

Comments

A primitive triangle is one for which the sides have no common factor.

A004611 gives the values of B, and A089025 gives the values of C.

Examples

			7 appears in the sequence because there exists a primitive 60-degree triangle with sides 7, 37 and 40.
		

Links

• Wikipedia, Integer triangle

Crossrefs

Cf. A004611, A089025, A229839.

Programs

PARI

\\ Gives terms not exceeding amax
\\ e.g. pt60a(25) gives [3,5,7,8,9,11,13,15,16,17,19,21,23,24,25]
pt60a(amax) = {
  s=[];
  for(m=1, amax\2,
    for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        if(2*m*n+n*n<=amax, s=concat(s, 2*m*n+n*n));
        if(m*m-n*n<=amax, s=concat(s, m*m-n*n))
      )
    )
  );
  vecsort(s,,8)
}

Formula

Empirical g.f.: -x*(x^5-x^4-x^3-2*x^2-2*x-3) / ((x-1)^2*(x^4+x^3+x^2+x+1)).