A229858 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of A.

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Colin Barker, Oct 06 2013

Keywords

• nonn
• easy

Comments

A229859 gives the values of B, and A050931 gives the values of C.

This sequence contains every integer larger than 8. - Nathaniel Johnston, Oct 06 2013

Examples

			12 appears in the sequence because there exists a 120-degree triangle with sides 12, 20 and 28.
		

Links

• Wikipedia, Integer triangle
• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (2, -1).

Crossrefs

Cf. A050931, A229849, A229859.

Programs

PARI

\\ Gives values of A not exceeding amax.
\\ e.g. t120a(20) gives [3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
t120a(amax) = {
  v=pt120a(amax);
  s=[];
  for(i=1, #v,
    for(m=1, amax\v[i],
      if(v[i]*m<=amax, s=concat(s, v[i]*m))
    )
  );
  vecsort(s,,8)
}
\\ Gives values of A not exceeding amax in primitive triangles.
\\ e.g. pt120a(20) gives [3, 5, 7, 9, 11, 13, 15, 16, 17, 19]
pt120a(amax) = {
  s=[];
  for(m=1, (amax-1)\2,
    for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        a=m*m-n*n;
        b=n*(2*m+n);
        if(a>b, a=b);
        if(a<=amax, s=concat(s, a))
      )
    )
  );
  vecsort(s,,8)
}

Formula

a(n) = n+4 for n>4.

a(n) = 2*a(n-1)-a(n-2) for n>6.

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