Relation to square roots times a constant greater than 1

"Relation to square roots times a constant greater than 1"
(*start*)
a = 1/10;(*"a" is a fraction of the form 1/positiveInteger*)
b = 0;
nn = 64;
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[-(a^(-1) - 1) + Sqrt[2/a*(n - 1 - (1 - a))]], {n, 2, nn}]
Show[ListLinePlot[t1],
 ListLinePlot[t2]]
(*end*)


"Relation to square roots times a constant a = 1 upperbound + 1"
(*start*)
a = 1/1;(*"a" is a fraction of the form 1/positiveInteger*)
b = 0;
c = 1;
nn = 20;

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), 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[-(a^(-1) - 1) - 1 +
    Sqrt[-1/1 + 2*2/a*((n - 1 + 1) - 1 - (1 - a))]], {n, 2, nn}]
Show[ListLinePlot[t1], ListLinePlot[t2]]
Clear[a, n]
Expand[-(a^(-1) - 1) - 1 +
  Sqrt[-1/1 + 2*2/a*((n - 1 + 1) - 1 - (1 - a))]]
(*end*)



"Relation to square roots times a constant a = 2 upperbound + 2"
(*start*)
a = 2/1;(*"a" is a fraction of the form 1/positiveInteger*)b = 0;
nn = 20;
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), 2 + 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[-(a^(-1) - 1) - 1 - 1/2 +
    Sqrt[-1/2 + 3*2/a*((n - 1) - 1 - (1 - a))]], {n, 2, nn}]
Show[ListLinePlot[t1], ListLinePlot[t2]]
Clear[a, n]
Expand[-(a^(-1) - 1) - 1 - 1/2 +
  Sqrt[-1/2 + 3*2/a*((n - 1) - 1 - (1 - a))]]
(*end*)


"Relation to square roots times a constant a = 3 upperbound + 3"
(*start*)
a = 3/1;(*"a" is a fraction of the form 1/positiveInteger*)b = 0;
c = 3;
nn = 80;
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), c + 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[-(1/a - 1) - (-1 + c - 1/c) +
    Sqrt[-1/c + (c + 1)*2/a*((n - 1 - 1) - 1 - (1 - a))]], {n, 2, nn}]
Show[ListLinePlot[t1, PlotStyle -> Red], ListLinePlot[t2]]
Clear[a, n];
Expand[-(1/a - 1) - (-1 + c - 1/c) +
  Sqrt[-1/c + (c + 1)*2/a*((n - 1 - 1) - 1 - (1 - a))]]
(*end*)