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

1,1

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

Links

Table of n, a(n) for n=1..66.

Wikipedia, Integer triangle

Kival Ngaokrajang, Illustration of initial terms

Index entries for linear recurrences with constant coefficients, signature (2, -1).

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.

Example

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

Prog

(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)
}

Crossrefs

Cf. A050931, A229849, A229859.

Sequence in context: A324060 A323828 A079581 * A269020 A057716 A138591

Adjacent sequences: A229855 A229856 A229857 * A229859 A229860 A229861

Keyword

nonn,easy

Author

Colin Barker, Oct 06 2013

Status

approved