Odd Squares, 8 (2^3), and the Sum of the First n Positive Integers

I did see it noted that the odd squares are 1 (mod 8) (http://mathworld.wolfram.com/SquareNumber.html). I imagine it is noted elsewhere, but they also have a relationship to the sum of the first n positive integers.

If it is noted that the sum of the first n positive integers = n(n+1)/2, then

((2^3)(((n(n+1))/2))+1=(2n+1)^2

(8(n^2+n)/2)+1=4(n^2)+2n+2n+1
(4(n^2+n))+1=4(n^2)+4n+1
4(n^2)+4n+1=4(n^2)+4n+1

for example:

the sum of the first positive integer, 1, is 1
then, (8*1)+1=9 or ((2*1)+1)^2=3^2

the sum of the first two positive integers, 1+2, is 3
then, (8*3)+1=25 or ((2*2)+1)^2=5^2

the sum of the first three positive integers, 1+2+3, is 6
then, (8*6)+1=49 or ((2*3)+1)^2=7^2

the sum of the first four positive integers, 1+2+3+4, is 10
then, (8*10)+1=81 or ((2*4)+1)^2=9^2

the sum of the first five positive integers, 1+2+3+4+5, is 15
then, (8*15)+1=121 or ((2*5)+1)^2=11^2

etc.