A308158 - OEIS

%I #11 Oct 20 2021 15:45:59

%S 0,0,1,0,1,2,3,2,4,5,7,6,8,10,13,11,14,17,20,18,22,25,29,27,31,35,40,

%T 37,42,47,52,49,55,60,66,63,69,75,82,78,85,92,99,95,103,110,118,114,

%U 122,130,139,134,143,152,161,156,166,175,185,180,190,200,211

%N Sum of the smallest side lengths of all integer-sided isosceles 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/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * ([i=k] + [i=n-i-k] - [k=n-i-k]) * k, where [] is the Iverson bracket.

%F Conjectures from _Colin Barker_, May 15 2019: (Start)

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

%F a(n) = a(n-3) + 2*a(n-4) - 2*a(n-7) - a(n-8) + a(n-11) for n>11.

%F (End)

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

%Y Cf. A059169.

%K nonn

%O 1,6

%A _Wesley Ivan Hurt_, May 14 2019