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Sum of the smallest side lengths of all integer-sided scalene triangles with perimeter n.
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%I #14 Sep 12 2021 21:22:58

%S 0,0,0,0,0,0,0,0,2,0,2,3,5,3,9,7,13,12,18,17,29,23,35,36,48,43,63,58,

%T 78,75,95,92,122,111,141,141,171,162,204,195,237,231,273,267,323,306,

%U 362,360,416,402,474,460,532,522,594,584,674,650,740,735,825

%N Sum of the smallest side lengths of all integer-sided scalene triangles with perimeter n.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>

%F a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} sign(floor((i+k)/(n-i-k+1))) * k.

%F Conjectures from _Colin Barker_, May 13 2019: (Start)

%F G.f.: x^9*(2 + 2*x + 2*x^2 + x^3) / ((1 - x)^4*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).

%F a(n) = -a(n-1) + 2*a(n-3) + 4*a(n-4) + 2*a(n-5) - a(n-6) - 5*a(n-7) - 5*a(n-8) - a(n-9) + 2*a(n-10) + 4*a(n-11) + 2*a(n-12) - a(n-14) - a(n-15) for n>15.

%F (End)

%t Table[Sum[Sum[k*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k + 1, Floor[(n - k - 1)/2]}], {k, Floor[(n - 1)/3]}], {n, 100}]

%o (PARI) a(n) = sum(k=1, (n-1)\3, sum(i=k+1, (n-k-1)\2, k*sign((i+k)\(n-i-k+1)))); \\ _Michel Marcus_, May 13 2019

%Y Cf. A307686.

%K nonn

%O 1,9

%A _Wesley Ivan Hurt_, May 13 2019