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A229838
Consider all primitive 60-degree triangles with sides A < B < C. The sequence gives the values of A.
1
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.
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
KEYWORD
nonn
AUTHOR
Colin Barker, Oct 01 2013
STATUS
approved