Number Spiral: Edges

$$\begin{array}{rrrrrr}1& 2& 9& 10& 25& \dots \\ 4& 3& 8& 11& 24& \dots \\ 5& 6& 7& 12& 23& \dots \\ 16& 15& 14& 13& 22& \dots \\ 17& 18& 19& 20& 21& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮\end{array}$$

Let $f(y,x)$ represent the number in row $y$ and column $x$.

Top Edge

$$f(1,x)=\{\begin{array}{ll}{x}^2& \text{ if }x\text{ is odd},\\ (x-1{)}^2+1& \text{ if }x\text{ is even}.\end{array}$$

Left Edge

$$f(y,1)=\{\begin{array}{ll}{y}^2& \text{ if }y\text{ is even},\\ (y-1{)}^2+1& \text{ if }y\text{ is odd}.\end{array}$$

Places Relative to Diagonal

$(y,x)$ falls exactly on the diagonal if and only if $y=x$.

$(y,x)$ falls above/on the right of diagonal if and only if $y<x$.

$(y,x)$ falls below/on the left of the diagonal if and only if $y>x$.

Formulas

$$f(y,x)=\{\begin{array}{ll}f(1,x)-(y-1)& \text{ if }y\le x\text{ and }x\text{ is odd},\\ f(1,x)+(y-1)& \text{ if }y\le x\text{ and }x\text{ is even},\\ f(y,1)+(x-1)& \text{ if }y\ge x\text{ and }x\text{ is odd},\\ f(y,1)-(x-1)& \text{ if }y\ge x\text{ and }x\text{ is even}.\end{array}$$