Detour, 20 bytes

This one is going for the "infinite sequence" option.

v1vq:$
  $+
p,p^
^ q

Try it online!

Branch 1 takes a number, prints it, adds it with the number from Branch 2, then puts the result in Branch 2
Branch 2 takes a number, feeds it to the addition with branch 1 then puts the original number (not the sum) in Branch 1.

For a better explanation click the link and you'll see it in action.

More "readable" version:

Detour, 267 bytes

:$v  1v   q   # split into branches
         
          +   # push sum of last 2 fibonacci numbers to branch 2
      {  

  p  , p   ^   # print branch 1, merge with branch 3

      }

  ^   q       # push branch 2 into branch 1 for printing and recycling

# 1   2   3

Try it online!

Detour, 20 bytes

This one is going for the "infinite sequence" option.

v1vq:$
  $+
p,p^
^ q

Try it online!

Branch 1 takes a number, prints it, adds it with the number from Branch 2, then puts the result in Branch 2
Branch 2 takes a number, feeds it to the addition with branch 1 then puts the original number (not the sum) in Branch 1.

For a better explanation click the link and you'll see it in action.

More "readable" version:

Detour, 267 bytes

:$v  1v   q   # split into branches
         
          +   # push sum of last 2 fibonacci numbers to branch 2
      {  

  p  ,p   ^   # print branch 1, merge with branch 3

      }

  ^   q       # push branch 2 into branch 1 for printing and recycling

# 1   2   3

:$v  1v   q   # split into branches
         
          +   # push sum of last 2 fibonacci numbers to branch 2
      {  

  p  ,p   ^   # print branch 1, merge with branch 3

      }

  ^   q       # push branch 2 into branch 1 for printing and recycling

# 1   2   3

Try it online!

Detour, 20 bytes

This one is going for the "infinite sequence" option.

v1vq:$
  $+
p,p^
^ q

Try it online!

Branch 1 takes a number, prints it, adds it with the number from Branch 2, then puts the result in Branch 2
Branch 2 takes a number, feeds it to the addition with A,branch 1 then puts the original number (not the sum) in Branch 1.

For a better explanation click the link and you'll see it in action.

More "readable" version:

Detour, 267 bytes

:$v  1v   q   # split into branches
         
          +   # push sum of last 2 fibonacci numbers to branch 2
      {  

  p  ,p   ^   # print branch 1, merge with branch 3

      }

  ^   q       # push branch 2 into branch 1 for printing and recycling

# 1   2   3

Try it online!