#Python 3.8, 104 74 65 60 57 bytes

lambda n:(-2,6)[n>4*(x:=(n**.5+1)//2)*x+2*x]*x+2+~n+8*x*x

Edit: Thanks to Johnathan Allan for getting it from 74 to 57 bytes!

This solution uses 0-based indexing.

Python 3.8, 104 74 65 60 57 bytes

lambda n:(-2,6)[n>4*(x:=(n**.5+1)//2)*x+2*x]*x+2+~n+8*x*x

Edit: Thanks to Johnathan Allan for getting it from 74 to 57 bytes!

This solution uses 0-based indexing.

#Python 3, 104 74 65 bytes

def f(n):x=((n-1)**.5+1)//2;return(-2,6)[n>4*x*x+2*x]*x+2-n+8*x*x

Edit: Thanks to Johnathan Allan for getting it from 74 to 65 bytes!

#Python 3.8, 104 74 65 60 57 bytes

lambda n:(-2,6)[n>4*(x:=(n**.5+1)//2)*x+2*x]*x+2+~n+8*x*x

Edit: Thanks to Johnathan Allan for getting it from 74 to 57 bytes!

This solution uses 0-based indexing.

#Python 3, 104 74 bytes

def a(n):x=(int((n-1)**.5)+1)//2;return 8*x**2+(6,-2)[n<=4*x**2+2*x]*x+2-n

#Python 3, 104 74 65 bytes

def f(n):x=((n-1)**.5+1)//2;return(-2,6)[n>4*x*x+2*x]*x+2-n+8*x*x

Edit: Thanks to Johnathan Allan for getting it from 74 to 65 bytes!