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

1,1

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.

Links

Table of n, a(n) for n=1..65.

Wikipedia, Integer triangle

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)).

Example

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

Prog

(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)
}

Crossrefs

Cf. A004611, A089025, A229839.

Sequence in context: A184996 A153309 A047486 * A191257 A120212 A093670

Adjacent sequences: A229835 A229836 A229837 * A229839 A229840 A229841

Keyword

nonn

Author

Colin Barker, Oct 01 2013

Status

approved