Why do we get this pattern (note what has been highlighted or italicized) when you divide consecutive square numbers with consecutive odd numbers ?
4 ÷ 3 = 1 R 1
9 ÷ 5 = 1 R 4
16 ÷ 7 = 2 R 2
25 ÷ 9 = 2 R 7
36 ÷ 11 = 3 R 3
49 ÷ 13 = 3 R 10
64 ÷ 15 = 4 R 4
81 ÷ 17 = 4 R 13
100 ÷ 19 = 5 R 5
121 ÷ 21 = 5 R 16
144 ÷ 23 = 6 R 6
169 ÷ 25 = 6 R 19
196 ÷ 27 = 7 R 7
225 ÷ 29 = 7 R 22
256 ÷ 31 = 8 R 8
Your observation (in the bold) is that $(2n)^2$ divided by $4n-1$ has the result $n$ R $n$.
$(2n)^2=4n^2$ and if you multiply the divisor $(4n-1)$ by the quotient $n$ and add the remainder $n$ you get $((4n-1)\cdot n)+n=4n^2-n+n=4n^2$.
So that pattern is just what you expect.
Your observation (in italics) is that $(2n+1)^2$ divided by $4n+1$ has the result $n$ R $(3n+1)$.
$(2n+1)^2=4n^2+4n+1$ and if you multiply the divisor $(4n+1)$ by the quotient $n$ and add the remainder $3n+1$ you get $((4n+1)\cdot n)+3n+1=4n^2+n+3n+1=4n^2+4n+1$.
So that pattern is just what you expect.
