"Relation to square roots"
(*start*)
a = 1;(*"a" is a positive small integer*)
b = 0;
nn = 100;
TableForm[
L2 = Table[
LinearProgramming[
Table[1/n, {n, 1, k}], {Table[If[n == 1, k, 1], {n, 1, k}]}, {{1,
0}}, Table[
If[n == 1, {-1, 1}, {-a*(n - 1), b*(n - 1)}], {n, 1, k}]], {k,
1, nn}]];
t1 = Table[Sum[If[L2[[n, k]] <= -1, 1, 0], {k, 2, n}], {n, 2, nn}]
t2 = Table[Round[Sqrt[2/a*(n - 1)]], {n, 2, nn}]
t2 - t1
Count[%, 0] - nn + 1
(*end*)