$$$2*f(n-1)$$$ comes from the fact that the salesman can start anywhere in the first column and then connect it to the $$$n-1$$$ previous columns.

$$$4*f(n-2)$$$ denotes the zig-zag type starting from the first column that we missed previously.

$$$2^n$$$ denotes the ways that start with the first column and end at the first column.

$$$2*t(n-1)$$$ is self explanatory. That just tells that for all the ways in $$$t(n-1)$$$ when the sales person comes to $$$n-1^{th}$$$ column just come to $$$n^{th}$$$ column in $$$2$$$ ways.

$$$4*f(n-2)$$$: Now since $$$n-1^{th}$$$ column has become a middle column for the $$$n^{th}$$$ column we count their ways too. It's just casework to figure out why the coefficient is $$$4$$$.