N[(Sqrt[3])]

N[(Sqrt[200])]

Table[{k, Floor[(Sqrt[2 k]) + (1/2)]}, {k, 1, 200}]

a[1] = 1;  a[n_] := a[n] = a[n - a[n - 1]] + 1

a[3]

Table[a[n], {n, 1, 100}]