Calculating total slots

05AB1E, 22 bytes

v¯R¬yQiõˆ}2£yåiˆ}yˆ}¯g

Try it online or verify all test cases.

Explanation:

v           # Loop over the integers `y` of the (implicit) input-list:
 ¯R         #  Push the global_array, and reverse it
   ¬        #  Get the first item (without popping the reversed global_array itself)
    yQi  }  #  If it's equal to the integer `y`:
       õˆ   #   Add an empty string to the global_array
   2£       #  Then only leave the first 2 items of the reversed global_array
     yåi }  #  If the integer `y` is in these first 2 items:
        ˆ   #   Add the (implicit) input-list to the global_array
 yˆ         #  And push the integer `y` itself to the global_array
}¯g         # After the loop: push the global array, and then pop and push its length
            # (which is output implicitly as result)

Brachylog, 10 bytes

It's always nice to see problem where Brachylog performs best

⊆Is₃ᶠ≠ᵐ∧Il

Explanation

⊆I           # Find the minimal ordered superset of the input (and store in I) where:
   s₃ᶠ       #     each substring of length 3
      ≠ᵐ     #     has only distinct numbers
        ∧Il  # and output the length of that superset

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R, 123 bytes

`-`=nchar;x=scan(,'');while(x!=(y=gsub("([^,]+),(([^,]*,){0,1})\\1(,|$)","\\1,\\2,\\1\\4",x)))x=y;-gsub("[^,]","",y)+(-y>1)

Try it online - single program!

Try it online - multiple examples!

A full program that reads a comma-separated list of integers as the input, and outputs the slots needed. I’m sure this could be golfed some more, and implementing this regex-based solution in some other languages would be more efficient in bytes.

Note on the second TIO I’ve wrapped it in a function to permit multiple examples to be shown. This function also shows the final list, but this is not output my the main program if run in isolation.